Explanation:
Okay, so I need to figure out how many ways 7 people can sit around a round table, but with the condition that Pierre, Rosa, and Thomas (let me call them P, R, T for short) can't sit next to each other. Also, rotations of the same arrangement are considered the same, so it's a circular permutation problem. Alright, let me start by recalling some basics about circular permutations.

First, for circular permutations, the number of ways to arrange n people around a round table is (n-1)! because rotations are considered the same. So, if there were no restrictions, the total number of arrangements would be (7-1)! = 6! = 720. But here, we have restrictions: P, R, and T can't sit next to each other. So, I need to subtract the arrangements where at least two of them are sitting together. Wait, but inclusion-exclusion might be necessary here. Hmm.

Alternatively, maybe it's easier to first arrange the other people and then place P, R, T in the gaps? Let me think. That's a common technique for problems where certain people can't be adjacent. Let me try that approach.

So, there are 7 people total. Let's consider the other 4 people (not P, R, T) first. Let me call them A, B, C, D. If we fix the positions of A, B, C, D around the table, then we can place P, R, T in the gaps between them. However, since it's a circular table, the number of gaps between the 4 people would be equal to 4. Wait, but arranging 4 people around a circular table: normally, the number of arrangements is (4-1)! = 6. But hold on, if we fix the positions of A, B, C, D, then the number of arrangements is actually (4-1)! = 6? Wait, no, if we fix one person to account for circularity, then the rest can be arranged linearly. Wait, maybe I need to think more carefully.

Wait, actually, when arranging people around a circular table, fixing one person's position removes the rotational symmetry. So, if I fix one person, say A, then arrange the remaining 3 people around the table. But perhaps I need to consider arranging A, B, C, D first. However, since we need to place P, R, T in the gaps between them, maybe it's better to first seat the 4 non-P-R-T people, then place P, R, T in the gaps.

But in circular permutations, when we arrange the 4 people, the number of ways is (4-1)! = 6. Then, once they are seated, there are 4 gaps between them where we can place the other people. So, if we have 4 gaps, and we need to place 3 people (P, R, T) into these gaps such that no two are in the same gap (since if two are in the same gap, they would be adjacent). Wait, but since the table is circular, arranging the 4 people creates 4 gaps between them. Each gap is between two people.

Therefore, if I want to place P, R, T into these gaps, such that each is in a separate gap, then the number of ways would be: first, arrange the 4 people around the table (which is (4-1)! = 6 ways), then choose 3 gaps out of the 4, and arrange P, R, T in those gaps. So, the number of ways would be 6 * C(4,3) * 3!.

Calculating that: 6 * 4 * 6 = 6 * 24 = 144? Wait, C(4,3) is 4, and then 3! is 6, so 4*6=24, times 6 is 144. So, total number of arrangements would be 144? But wait, the total number of unrestricted arrangements is 720. So, 144 is less than that. Wait, but is this the correct approach?

Wait, but hold on. If we first fix the positions of the 4 people around the table, then arrange the others. However, in circular permutations, when we fix one person, we're already accounting for rotational symmetry. But if we first fix the 4 people, then the number of ways is (4-1)! as before. But then placing P, R, T into gaps. Wait, but another thought: when placing P, R, T into gaps, since the table is circular, once we fix the positions of the 4 people, the gaps are fixed. Therefore, the number of ways to place P, R, T into the gaps without them sitting together is C(4,3)*3!.

But is this correct? Let me think again. If there are 4 gaps, each gap can hold any number of people, but if we want to prevent any two of P, R, T from sitting next to each other, we need each of them in separate gaps. Therefore, yes, we have to choose 3 gaps out of 4, and then arrange P, R, T in those gaps. Each such arrangement would result in P, R, T not sitting next to each other. Therefore, the total number is indeed (number of ways to arrange the 4 people) * (number of ways to choose gaps) * (number of ways to arrange P, R, T in those gaps).

But let me verify this approach with an example. Suppose we have 4 people around the table: A, B, C, D. Let me fix their positions. For example, fix A at a position, then B, C, D can be arranged in 3! = 6 ways. Then there are 4 gaps: between A & B, B & C, C & D, D & A. So, if we choose any 3 of these gaps, say the first, second, and third gaps, and place P, R, T in them. Then, each of P, R, T is separated by at least one person. Therefore, they are not adjacent. Therefore, this approach should work.

But then, the total number is 6 * 4 * 6 = 144. However, let me check if there's another approach. Alternatively, maybe using inclusion-exclusion. Let me try that as well to see if I get the same answer.

Total number of circular arrangements: (7-1)! = 720.

Now, subtract the arrangements where at least two of P, R, T are sitting together. Wait, but inclusion-exclusion can get complicated here. Let me try.

First, compute the number of arrangements where at least one pair of P, R, T are sitting together. Let me denote the three people as P, R, T. Let me define:

A: arrangements where P and R are sitting together.

B: arrangements where P and T are sitting together.

C: arrangements where R and T are sitting together.

We need to compute |A ∪ B ∪ C|, and subtract that from the total to get the number of arrangements where none of the pairs are sitting together.

By inclusion-exclusion:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, first compute |A|. How many circular arrangements have P and R sitting together?

To compute |A|, treat P and R as a single entity. Then, we have 6 entities to arrange around the table: the PR pair, T, A, B, C, D. However, since it's circular, the number of arrangements is (6-1)! = 120. But also, the PR pair can be in two orders: PR or RP. So, |A| = 2 * 5! = 2 * 120 = 240? Wait, 5! is 120, so 2 * 120 = 240? Wait, but hold on: Wait, the formula for circular permutations when we have a pair is (n-1)! for the number of arrangements of the entities, times the internal arrangements. So yes, for each pair, it's 2*(n-2)! ?

Wait, let me think. If we have n people, and we consider two people as a single entity, then the number of circular arrangements is (n-1)! for the entities, but since the two people can be arranged in 2 ways, it's 2*(n-1)!? Wait, but actually, when you have a pair, the number of entities is (n - 1), because we merged two into one. Therefore, the number of circular arrangements would be (n - 2)! * 2? Wait, no:

Wait, the standard formula for circular permutations with some people together is: treat the group as one entity, so total entities = n - 1. Then the number of circular arrangements is (n - 2)! * 2 (for the internal arrangement). Wait, let me check with n=3: if we have 3 people, and want to count arrangements where two are together. Treating them as a pair, then number of circular arrangements is (2-1)! * 2 = 1 * 2 = 2. But in reality, for 3 people around a table, there are (3-1)! = 2 arrangements. However, if we consider two people fixed together, then there's only 1 way to arrange them (since rotations are considered same). Wait, this is confusing.

Wait, maybe I need to think differently. Let me take a concrete example. Suppose n=3 people: A, B, C. How many circular arrangements where A and B are together? Well, in circular arrangements, if we fix A's position, then B can be to the left or right of A, and C is fixed. So, there are 2 arrangements. But according to the formula (n - 2)! * 2: n=3, so (3-2)! * 2 = 1 * 2 = 2. Correct. But if we didn't fix positions, the total number of circular arrangements for 3 people is 2. So, yes, seems like the formula works. So in general, for circular arrangements where a specific pair is together, it's 2*(n - 2)! ?

Wait, but in our case, n=7, so for |A|, the number of arrangements where P and R are together is 2*(7 - 2)! = 2*5! = 240? Wait, but hold on, when we merge P and R into a single entity, we have 6 entities: the PR pair, T, A, B, C, D. Then, the number of circular arrangements is (6 - 1)! = 5! = 120. Multiply by 2 for the internal arrangement of PR, so 240. So yes, that's |A|.

Similarly, |B| and |C| are the same. So, each of |A|, |B|, |C| is 240. Therefore, |A| + |B| + |C| = 3 * 240 = 720.

But wait, that's equal to the total number of circular arrangements? That seems odd. Because 720 is the total number of arrangements. But if we subtract |A ∪ B ∪ C| from total arrangements, which would be 720 - |A ∪ B ∪ C|, but if |A ∪ B ∪ C| is 720, that would imply zero, which can't be. So, clearly, something is wrong here. Wait, but |A| is 240, but if we have three such sets, their total is 720, but they have overlaps. So, we need to subtract the intersections. So, let me continue.

Now, |A ∩ B|: arrangements where both P and R are together, and P and T are together. Wait, if P is with both R and T, that means P is between R and T, so effectively, the three of them form a block. Similarly, |A ∩ B| would be the number of arrangements where P is adjacent to both R and T, meaning that P, R, T are all sitting together in a block. Similarly for other intersections. Let me confirm.

Yes, because if P and R are together, and P and T are together, then R and T must also be adjacent to P, so all three are in a block. Therefore, |A ∩ B| is equal to the number of circular arrangements where P, R, T are all together. Similarly, |A ∩ C| would be the same: arrangements where P and R are together and R and T are together, which again implies all three are together. Similarly |B ∩ C| is same. So each |A ∩ B|, |A ∩ C|, |B ∩ C| is equal to the number of arrangements where P, R, T are all together.

So, let's compute that. How many circular arrangements have P, R, T all together. Treat them as a single block. Then, we have 5 entities: the PRT block, A, B, C, D. Number of circular arrangements is (5-1)! = 4! = 24. Then, the block PRT can be arranged in 3! = 6 ways. Therefore, total |A ∩ B| = 6 * 4! = 6 * 24 = 144. Wait, 4! is 24? Wait, no. Wait, (5-1)! is 4! = 24. Multiply by 3! for arranging the block. So 24 * 6 = 144. But each intersection |A ∩ B|, |A ∩ C|, |B ∩ C| is 144. Therefore, the total |A ∩ B| + |A ∩ C| + |B ∩ C| = 3 * 144 = 432.

Now, the inclusion-exclusion formula says:

|A ∪ B ∪ C| = 720 - 432 + |A ∩ B ∩ C|.

Wait, no. Wait, the formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, we have:

|A ∪ B ∪ C| = 720 - 432 + |A ∩ B ∩ C|.

Wait, no. Wait, original formula:

|A ∪ B ∪ C| = (sum |A|) - (sum |A ∩ B|) + |A ∩ B ∩ C|.

So substituting numbers:

= 720 - 432 + |A ∩ B ∩ C|.

But we need to compute |A ∩ B ∩ C|. This is the number of arrangements where P and R are together, P and T are together, and R and T are together. Which is exactly the same as all three being together. So, |A ∩ B ∩ C| is same as the number of arrangements where all three are together, which is same as before. Wait, but in the previous calculation, when we calculated |A ∩ B|, we considered that as all three together. So, |A ∩ B ∩ C| is also the same, because if all three pairs are together, then all three must be together. Therefore, |A ∩ B ∩ C| is same as the number of arrangements where all three are together, which is 144? Wait, no. Wait, no. Wait, when we calculated |A ∩ B|, we treated the block of PRT as one entity, leading to 4! * 3! = 24 * 6 = 144. However, in reality, if all three are together, then the number of circular arrangements is (number of ways to arrange the block and other people) * (internal arrangements). Wait, but if I have 5 entities: the PRT block, A, B, C, D. Then, circular arrangements: (5 - 1)! * 3! = 4! * 6 = 24 * 6 = 144. So yes, same as before.

But that seems like |A ∩ B ∩ C| is also 144? Wait, but if we have all three pairs together, that's the same as having all three together. So |A ∩ B ∩ C| is 144. Therefore, putting it back into inclusion-exclusion:

|A ∪ B ∪ C| = 720 - 432 + 144 = 720 - 432 is 288; 288 + 144 = 432. Wait, so |A ∪ B ∪ C| is 432? Therefore, the number of arrangements where at least one pair of P, R, T are sitting together is 432. Therefore, the number of arrangements where none of them are sitting together is total arrangements minus that, which is 720 - 432 = 288. Wait, but earlier approach using gaps gave me 144. So which is correct?

This discrepancy is concerning. Which one is wrong?

Let me check with small numbers. Let me take a smaller case where I can compute manually. Let me try with n=4 people: P, R, T, and one more person, say A. Let's say we want to find the number of circular arrangements where P, R, T are not sitting together. Wait, but in n=4, there are only 4 seats. If P, R, T are three people, they can't all be separated because there's only 4 seats. Wait, so for n=4, the number of arrangements where no two of P, R, T are adjacent? Let's see. Since there are only 4 seats, if we seat P, R, T, and A. To have no two of P, R, T adjacent, they need to be placed with at least one person between them. But with 4 seats, and 3 people, it's impossible because if you seat them with one seat between each, you need 3 seats for the people and 1 seat between them, but in a circle, that would require 3*2=6 seats? No, wait. For circular arrangements, the number of seats is equal to the number of people. So with 4 seats, how to place 3 people such that none are adjacent? Wait, in a circular table with 4 seats, each seat has two neighbors. If we place 3 people, then at least two of them must be adjacent because if you have 3 people, they occupy 3 seats, leaving 1 seat. The 1 seat is between two people, so those two people are adjacent. Therefore, it's impossible. Therefore, the answer is zero. Let me check with both methods.

First method: using the gap method. Number of ways to arrange the 1 non-P-R-T person (A) around the table: (1-1)! = 1? Wait, but arranging 1 person around a circular table is 0!? Wait, no. Wait, the formula is (n-1)! for n people. So if we have n=1, the number of circular arrangements is 0!? Wait, actually, for n=1, the number of circular arrangements is 1 (just the person sitting alone). So (1-1)! = 0! = 1. So arranging A around the table: 1 way. Then, the number of gaps is equal to 1. Wait, if there is only one person, there is one gap between them (but since it's a circle, it's just one gap). Wait, but we need to place 3 people into these gaps, but there's only 1 gap. Therefore, we can't place them all into separate gaps. Therefore, the number of ways would be zero. Which matches the actual answer. So the gap method gives zero.

Now, using inclusion-exclusion. Total arrangements: (4-1)! = 6. Number of arrangements where at least one pair of P, R, T are adjacent. Let me compute that. Using inclusion-exclusion.

First, |A|, |B|, |C| each: treating a pair as a block. So for |A|: number of arrangements where P and R are together. So treat them as a block, so we have 3 entities: PR, T, A. Number of circular arrangements: (3-1)! = 2. Multiply by 2 for internal arrangements: 2*2=4. Similarly for |B| and |C|. So total |A| + |B| + |C| = 3*4=12. Then, |A ∩ B|, which is arrangements where P and R are together and P and T are together, which implies all three are together. Number of arrangements: treat PRT as a block, so entities are PRT and A. Number of circular arrangements: (2-1)! = 1. Multiply by internal arrangements: 3! = 6. So total |A ∩ B| = 6. Similarly for |A ∩ C| and |B ∩ C|, so total is 3*6=18. Then |A ∩ B ∩ C| is same as all three together, which is 6. Therefore, inclusion-exclusion formula:

|A ∪ B ∪ C| = 12 - 18 + 6 = 0? Wait, that can't be. Wait:

Wait, formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So substituting numbers:

= 12 - 18 + 6 = 0? That would mean that there are no arrangements where at least one pair is together? Which is impossible, since we know that in all arrangements, at least two of P, R, T must be adjacent. Therefore, there must be a mistake in calculations.

Wait, let me recalculate. For n=4, total arrangements is 6. How many of them have at least one pair of P, R, T adjacent? Let's list them.

Since n=4, total circular permutations is 3! = 6. Let me fix A's position to remove rotational symmetry. Let me list all possible arrangements:

Let's fix A at a position. Then, the remaining 3 people are P, R, T.

Possible arrangements:

1. P, R, T, A

2. P, T, R, A

3. R, P, T, A

4. R, T, P, A

5. T, P, R, A

6. T, R, P, A

But since it's a circle, some of these are rotations. Wait, no, since we fixed A's position, these are all distinct.

Wait, but actually, when we fix A's position, the number of arrangements is 3! = 6. Let me check which of these have at least two of P, R, T adjacent.

Arrangement 1: P, R, T, A. So P and R are adjacent, R and T are adjacent, T and A are adjacent. So yes, multiple adjacents.

Arrangement 2: P, T, R, A. P and T are adjacent, T and R are adjacent, R and A are adjacent. Also multiple adjacents.

Arrangement 3: R, P, T, A. R and P adjacent, P and T adjacent, T and A adjacent.

Arrangement 4: R, T, P, A. R and T adjacent, T and P adjacent, P and A adjacent.

Arrangement 5: T, P, R, A. T and P adjacent, P and R adjacent, R and A adjacent.

Arrangement 6: T, R, P, A. T and R adjacent, R and P adjacent, P and A adjacent.

So all 6 arrangements have at least two of P, R, T adjacent. Therefore, |A ∪ B ∪ C| = 6. But according to inclusion-exclusion earlier, I had 0. Which is wrong. Therefore, there's a mistake in my calculation. Let me check again.

So for n=4, |A|: arrangements where P and R are together. How many?

Treating P and R as a block. So the block PR, T, A. Number of circular arrangements: since we fixed A's position, maybe the formula is different?

Wait, earlier I assumed that treating a pair as a block would give (number of entities - 1)! * internal arrangements. But if we fixed A's position, then the number of arrangements where P and R are together would be different?

Wait, maybe I need to adjust how I compute |A| in the inclusion-exclusion.

Alternatively, perhaps my formula for |A| was wrong. Let me think again.

In the case where we have n people, and we want the number of circular arrangements where a specific pair is together. If we fix one person's position, say A, then the total number of arrangements is (n-1)! So here, for n=4, fixing A, total arrangements is 3! = 6.

To compute |A| (P and R together): How many arrangements have P and R adjacent?

If we fix A's position, then we have three seats left: positions 1, 2, 3 (assuming A is fixed at position 4). Let me think of it as a line with fixed A at one end, but since it's a circle, fixing A's position makes it linear.

Wait, maybe it's easier to think linearly. When we fix A's position, the rest can be arranged linearly with the condition that P and R are adjacent.

Number of ways to arrange P, R, T in the remaining 3 seats with P and R adjacent. Let me think. In linear arrangements, with 3 seats, the number of arrangements where P and R are adjacent is 2 * 2! = 4. Wait:

Number of ways where P and R are adjacent: treat them as a block, which can be PR or RP. Then, the block and T can be arranged in 2 positions: block and T, or T and block. So 2 * 2 = 4. Then, since we fixed A's position, the total is 4. Therefore, |A| is 4. Similarly, |B| and |C| are each 4. Therefore, |A| + |B| + |C| = 12. However, since we fixed A's position, the total number of arrangements is 6. Therefore, clearly, my previous calculation was wrong in assuming that |A| is 2*(n - 2)! etc. Because that would have given |A| as 2*(4 - 2)! = 2*2 = 4, which matches here. Wait, but then why did my previous inclusion-exclusion give the wrong result?

Let me recast the inclusion-exclusion for n=4. So:

Total arrangements: 6.

|A| = |B| = |C| = 4.

Sum |A| + |B| + |C| = 12.

Now, |A ∩ B|: arrangements where P and R are together AND P and T are together. Which implies all three are together. How many such arrangements?

If P, R, T are together, then they form a block. With A fixed, we have the block and A. Since it's a circular table with A fixed, the block can be placed in only one position relative to A? Wait, no. Wait, if we fix A's position, then the block can be placed in positions 1-2-3, but since it's a circle, fixing A's position makes it linear? Wait, maybe not. Let me think again. If A is fixed at position 4, the other three positions are 1, 2, 3. If we have a block of P, R, T occupying positions 1, 2, 3 in some order, then there's only one block. But since the block can be arranged in 3! ways, but considering the circular table, is that correct? Wait, no, because A is fixed. So the block can be arranged in 3! ways, but since we fixed A's position, the number of arrangements where P, R, T are together is 3! = 6. But that can't be, since total arrangements is 6. But in reality, all arrangements have P, R, T together? Which is not true. Wait, for example, if we have arrangement P, T, R, A (which is arrangement 2), then P, T, R are in positions 1, 2, 3. But if we have another arrangement, say, P, T, A, R? Wait, no, since A is fixed at position 4, the positions are fixed. So the three seats are positions 1, 2, 3. So any arrangement of P, R, T in these positions would have them all together. Wait, but in that case, if you fix A at position 4, then the three seats are adjacent to each other (since it's a circle). So positions 1, 2, 3 are adjacent in a circle to position 4 (A). Therefore, positions 1, 2, 3 are adjacent in a circle to each other? No. Wait, in a circular table with 4 seats, seat 1 is adjacent to seat 2 and 4. Seat 2 is adjacent to 1, 3, and 4? No, seat 2 is adjacent to seat 1 and seat 3. Seat 3 is adjacent to seat 2 and seat 4. Seat 4 is adjacent to seat 3 and seat 1. So if A is fixed at seat 4, then seats 1, 2, 3 are arranged linearly around the table, with seat 1 adjacent to seat 4 and seat 2, seat 2 adjacent to seat 1 and seat 3, seat 3 adjacent to seat 2 and seat 4. Therefore, seats 1, 2, 3 are not all adjacent to each other. Seat 1 is only adjacent to 2 and 4. Seat 3 is only adjacent to 2 and 4. So if we place P, R, T in seats 1, 2, 3, they are not all adjacent to each other; only seat 1 and 2, and seat 2 and 3 are adjacent. Therefore, in this case, if we have P, R, T in seats 1, 2, 3, then P and R may be adjacent, R and T may be adjacent, but P and T are not adjacent unless they are in adjacent seats. Wait, but in the example where seats 1, 2, 3 have P, T, R: seat 1 is P, seat 2 is T, seat 3 is R. Then P and T are adjacent (seats 1 and 2), T and R are adjacent (seats 2 and 3), but P and R are not adjacent. Therefore, in this arrangement, only two pairs are adjacent. Therefore, if we have all three of P, R, T in seats 1, 2, 3, they are not all adjacent to each other, but two of them are.

Therefore, in this case, the number of arrangements where all three are adjacent would be zero? Wait, but there is no way to have all three adjacent, because to have all three adjacent, they would need to be in seats 1, 2, 3 with each adjacent to the next, but seat 1 and seat 3 are not adjacent. So in a circle of 4 seats, you cannot have three people all adjacent. Therefore, |A ∩ B| would be zero? Wait, but earlier calculation said that treating them as a block gives 3! = 6, but that's incorrect. So my mistake was in the assumption that treating them as a block would give valid arrangements. Actually, in this case, because of the circular table, when we fix A's position, you cannot have all three P, R, T together. Therefore, my formula was wrong.

This shows that my previous approach for inclusion-exclusion was incorrect. So how do I correctly calculate |A ∩ B|?

This suggests that my initial method for calculating |A ∩ B| was wrong. Therefore, going back to the original problem with n=7, my inclusion-exclusion approach might have been flawed. Therefore, maybe the gap method was correct?

Wait, going back to the original problem. If using the gap method, with 4 people seated, creating 4 gaps, placing 3 people into separate gaps, giving 6 * 4 * 6 = 144. But according to inclusion-exclusion with n=4, this method gives zero, which is correct. But with the inclusion-exclusion approach, I had conflicting results. Therefore, maybe the gap method is correct and the inclusion-exclusion needs to be carefully calculated.

Alternatively, maybe the problem arises because when we calculated |A ∩ B| in the original problem, I treated it as arrangements where all three are together, but in reality, there might be overcounting? Or undercounting?

Alternatively, perhaps the correct way is to use the gap method for the original problem, which gave 144, and then check with another approach.

But let's see. If for n=7, the answer is 144, but when using inclusion-exclusion with n=4, we saw that the gap method gives the correct answer. However, the inclusion-exclusion approach needs to be correctly calculated.

Alternatively, maybe I made a mistake in calculating |A ∩ B| in the original problem.

Let me try again for the original problem (n=7). Let me recast the inclusion-exclusion correctly.

Total number of circular arrangements: 6! = 720.

Now, |A|: arrangements where P and R are adjacent. Treat them as a block. So, number of entities: 6 (block, T, A, B, C, D). Number of circular arrangements: (6 - 1)! = 120. Multiply by 2 (for PR or RP): 240. So |A| = 240. Similarly |B| and |C| = 240 each. So sum is 720.

Now, |A ∩ B|: arrangements where P and R are adjacent AND P and T are adjacent. Which means that P is adjacent to both R and T, so P, R, T are all adjacent in a block. Therefore, treat P, R, T as a single block. Then, number of entities is 5: block, A, B, C, D. Number of circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of the block. How many ways to arrange P, R, T in the block? Since P is adjacent to both R and T, the block can be arranged as P-R-T, P-T-R, R-P-T, T-P-R, R-T-P, T-R-P. Wait, but if P must be adjacent to both R and T, then the three people can be arranged in the block in 2 * 2 = 4 ways? Wait, no. Let me think. If we have a block of P, R, T where P is adjacent to both R and T, then the block must be a sequence where P is between R and T. So, the block can be R-P-T or T-P-R. Therefore, two possibilities. Alternatively, if the block is a circle? Wait, no. The block is linear because it's part of a circular table. Wait, no, the block is part of the circular table. Wait, when we treat P, R, T as a block, the block itself is linear? Or can it be circular? Wait, no, because we are arranging them in a circular table. However, the block is just a linear block. Wait, for the purposes of counting, when we have a block of P, R, T, the internal arrangements are 3! = 6, since they can be in any order. However, in our case, the condition is that P is adjacent to both R and T. Therefore, the block must include P adjacent to both R and T, which would require that in the block, P is in the middle. So, for example, R-P-T or T-P-R. Therefore, only 2 possible arrangements for the block. Therefore, internal arrangements are 2, not 6. Therefore, |A ∩ B| is (number of circular arrangements of entities) * internal arrangements = 4! * 2 = 24 * 2 = 48? Wait, hold on. Wait, number of entities is 5 (block, A, B, C, D). Number of circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of the block: 2. Therefore, |A ∩ B| = 24 * 2 = 48. Similarly for |A ∩ C| and |B ∩ C|.

Wait, previously I had said |A ∩ B| was 144, which was incorrect because I forgot that the internal arrangements are not 6 but only 2. So earlier mistake was in assuming that the entire block of P, R, T can be arranged in 3! ways, but actually, when we are calculating |A ∩ B|, which is arrangements where P is adjacent to both R and T (i.e., P is between R and T), so the internal arrangements of the block are only 2, not 6. Therefore, the correct value of |A ∩ B| is 24 * 2 = 48. Similarly for the other intersections. Therefore, the total |A ∩ B| + |A ∩ C| + |B ∩ C| = 3 * 48 = 144.

Then, |A ∩ B ∩ C|: arrangements where P is adjacent to R, P is adjacent to T, and R is adjacent to T. So all three are adjacent to each other, meaning they form a block. Now, how many internal arrangements? Since all three are adjacent to each other, they can be arranged in any order around the block. However, since it's a circular table, the block itself is linear? Or does it matter?

Wait, when we have a block of P, R, T where all three are adjacent to each other, the block can be arranged in 3! = 6 ways. So, number of circular arrangements is (number of entities: 5, the block, A, B, C, D) circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of block: 6. Therefore, |A ∩ B ∩ C| = 24 * 6 = 144.

Wait, but earlier when calculating |A ∩ B|, we had only 2 arrangements for the block. But for |A ∩ B ∩ C|, we have all three adjacent, so the block can be arranged in 3! ways. So, that seems okay.

Therefore, going back to inclusion-exclusion formula:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| = 720 - 144 + 144 = 720? Wait, no:

Wait, original values:

|A| + |B| + |C| = 720

|A ∩ B| + |A ∩ C| + |B ∩ C| = 3 * 48 = 144

|A ∩ B ∩ C| = 144

Therefore,

|A ∪ B ∪ C| = 720 - 144 + 144 = 720? Which can't be, because that would imply that all arrangements have at least one pair, which is not true. For example, there might be arrangements where P, R, T are all separated.

But according to the gap method, there are 144 such arrangements. Therefore, there's a contradiction here. Therefore, clearly, there's still a mistake in my calculations.

Wait, let's recast the inclusion-exclusion with corrected numbers.

Let me compute again:

Total arrangements: 720

|A| = |B| = |C| = 240, so sum is 720.

Each |A ∩ B| = |A ∩ C| = |B ∩ C| = ?

Earlier, we said that when we have both P-R and P-T adjacent, which implies P is between R and T, so the block is R-P-T or T-P-R, so internal arrangements: 2. Hence, |A ∩ B| = (number of circular arrangements with block) * internal arrangements. The number of circular arrangements with the block is (number of entities - 1)! where entities are 5 (block + A, B, C, D). Hence (5 - 1)! = 24, multiplied by 2 gives 48. Therefore, each |A ∩ B| = 48, so total for all intersections is 3 * 48 = 144.

Then, |A ∩ B ∩ C|: arrangements where all three pairs are adjacent, meaning the three are in a block, which can be arranged in 3! = 6 ways. Number of circular arrangements with the block: (5 - 1)! = 24, multiplied by 6 gives 24 * 6 = 144. Therefore, the inclusion-exclusion formula:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. Wait, that can't be. Therefore, the inclusion-exclusion formula says that |A ∪ B ∪ C| = 720 - 144 + 144 = 720? Which suggests that all arrangements have at least one pair of P, R, T adjacent, which contradicts the gap method which gave 144 arrangements where none are adjacent. Therefore, there is a fundamental error here.

Wait, let's think again. Where is the error?

Perhaps in calculating |A ∩ B|. If |A ∩ B| is arrangements where both P-R and P-T are adjacent, which implies that P is adjacent to both R and T, so they form a block of size 3. However, in such case, the block can be arranged in 2 ways: R-P-T or T-P-R. But when we calculate the number of arrangements with the block, we have (number of entities) = 5 (the block, A, B, C, D). Number of circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of the block: 2. So 24 * 2 = 48. However, is that correct?

Wait, but if we have a block of 3 people, then when we consider circular arrangements, is the block treated as a single entity? Yes. So, the number of circular arrangements for the block and the other 4 people is (5 - 1)! = 24. Then multiply by the internal arrangements of the block. However, if the block is linear, then internal arrangements are 2. If the block is circular, it would be different. Wait, no. When we treat the block as a single entity in the circular arrangement, the block itself is linear, because it's part of the circular table. So the internal arrangements are linear. Therefore, 2 possibilities, as before.

However, if we consider that the block could be arranged in more ways, perhaps?

Wait, suppose the block is R-P-T. Then, if we rotate the entire table, does that count as the same arrangement? No, because we already accounted for circular permutations by fixing the positions. Wait, no, the formula (number of entities - 1)! accounts for rotational symmetry. Therefore, the calculation seems correct.

But then, if |A ∩ B| = 48, and similarly for others, and |A ∩ B ∩ C| = 144, then:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. Which implies that all arrangements have at least one pair, which conflicts with the gap method. Therefore, there must be another error.

Wait, what is |A ∩ B ∩ C|? It is arrangements where all three are adjacent, which we calculated as 144. However, if we calculate the number of arrangements where all three are adjacent, it's 24 * 6 = 144, as previously. But how many total arrangements are there where all three are adjacent? Let me calculate it via gap method. Wait, if all three are adjacent, it's different from the gap method. Wait, but if we use the inclusion-exclusion, and if the result is that all arrangements have at least one pair, then there are zero arrangements where none are adjacent, which contradicts the gap method.

Alternatively, maybe my calculation of |A ∩ B| is wrong. Wait, let me think of another way: how many arrangements have P and R adjacent, and P and T adjacent. Let me think of it as fixing P's position. Since it's a circular table, fix P's position to remove rotational symmetry. Then, the number of arrangements where P is adjacent to both R and T. If P is fixed, then R and T must be in the two seats adjacent to P. There are 2 seats next to P. So R and T must occupy these two seats, which can be arranged in 2! = 2 ways. Then, the other 4 people (A, B, C, D) can be arranged in the remaining 4 seats in 4! = 24 ways. Therefore, total number of arrangements where P is adjacent to both R and T is 2 * 24 = 48. Which matches the previous calculation of |A ∩ B| = 48. Therefore, this seems correct.

Therefore, with this, |A ∩ B| = 48, as calculated. Then, how does the inclusion-exclusion formula proceed.

With |A ∪ B ∪ C| = 720 - 144 + 144 = 720. Which would imply that all arrangements have at least one pair, which can't be. Therefore, there must be an error in the inclusion-exclusion formula.

Wait, maybe the formula is different. Let me recall inclusion-exclusion formula:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, substituting numbers:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. But that suggests that |A ∪ B ∪ C| = 720. But the total number of arrangements is 720, which would mean that every arrangement has at least one pair, but that's not true. So, where is the mistake?

Wait, no! Wait, the formula is |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|. So substituting:

= 720 - 144 + 144 = 720? No, wait:

It's 720 - 144 + 144 = 720? Wait, that would be 720 - 144 is 576, plus 144 is 720. Yes. So this suggests that |A ∪ B ∪ C| = 720, which would mean that every arrangement is counted in at least one of A, B, C. But this contradicts our gap method, which suggests that there are arrangements where none of the pairs are adjacent.

Therefore, there must be an error in the calculation of |A|, |B|, |C|. Let me check.

Earlier, I calculated |A| as 240. Let me verify this with the fixed P method. Fix P's position. Number of arrangements where P and R are adjacent: P is fixed, R can be on either side of P. So 2 choices for R. Then, the remaining 5 people (including T, A, B, C, D) can be arranged in the remaining 5 seats: 5! = 120. Therefore, total number is 2 * 120 = 240. This matches the previous calculation of |A| = 240.

Similarly for |B| and |C|. Therefore, total |A| + |B| + |C| = 720.

But if |A ∪ B ∪ C| = 720, that would mean every arrangement is in at least one of A, B, C. But that is not true. For example, consider an arrangement where P, R, T are all separated. How is this possible?

Wait, in a 7-seat table, if you have 3 people, it's possible to seat them such that none are adjacent. For example, seat them with at least one person between each pair. For example, if we seat them with two seats between each of them. Let's see, with 7 seats, placing 3 people with at least one seat between them. The minimum number of seats needed is 3 + 2 = 5 (each person needs at least one seat between them). Since 7 >= 5, it is possible. Therefore, there must be arrangements where none of the pairs are adjacent. Hence, the inclusion-exclusion must have an error.

Where is the error? Let me think about the calculation of |A ∩ B ∩ C|. Earlier, I said it's 144, but if every arrangement is in |A ∪ B ∪ C|, then |A ∩ B ∩ C| would be the number of arrangements where all three pairs are adjacent, which is the same as all three being adjacent. But how many are those?

Wait, earlier calculation said 144. Let me verify with fixing P's position. Fix P's position. Then, the number of arrangements where all three are adjacent: P, R, T are together. So, the block can be placed in the table. Since P is fixed, the block can be to the left or right of P. Wait, no. If P is fixed, the block of P, R, T can be placed in two directions: P-R-T or P-T-R. But since the table is circular, fixing P's position, the block can extend clockwise or counterclockwise. Wait, but since P is fixed, the block can be placed in the two adjacent seats. Wait, if P is fixed, then to have P, R, T all together, R and T must be in the two neighboring seats of P, but that would only allow the block to be P-R-T or P-T-R, but that would only cover the immediate neighbors of P. However, in a 7-seat table, there are more seats. Wait, actually, no. If we want all three P, R, T to be together, the block can be placed anywhere, not just adjacent to P. So if P is fixed, the block can be placed in different positions.

Wait, this seems complex. Let me think differently. Number of circular arrangements where P, R, T are all adjacent: treat them as a block, so 5 entities. Number of circular arrangements: (5 - 1)! = 24. Multiply by 3! for internal arrangements: 24 * 6 = 144. But if I fix P's position, how many such arrangements are there?

If P is fixed, the block can start at P's position. The block can be P-R-T or P-T-R, extending to the right or left. But since it's a circle, fixing P, the block can be placed in different ways. Wait, fixing P's position, the block can be placed in different positions. Let me think.

Wait, with P fixed, the block of P, R, T can be placed in the table. The length of the block is 3 seats. In a circular table with P fixed, the number of ways to place a block of 3 seats containing P is 7 - 3 = 4? Wait, no. For a circular table, the number of ways to place a block of k seats is n, but since rotations are accounted for by fixing P's position. Wait, this is getting too tangled.

Alternatively, let me use the formula: number of arrangements where all three are together is 144. If total arrangements is 720, and |A ∪ B ∪ C| is 720, then that would mean that all arrangements have at least one pair, which contradicts the gap method. Therefore, there must be a mistake in the inclusion-exclusion calculation.

Wait, perhaps the error is in the calculation of |A ∩ B|. Wait, earlier, when I fixed P's position and calculated |A ∩ B| as 48, but if I use the formula for |A ∩ B| as 2 * (n - 2)! = 2 * 5! = 240? No, that doesn't make sense.

Wait, let me think about the inclusion-exclusion formula again. If |A ∪ B ∪ C| = 720, and the total number of arrangements is 720, then |A ∪ B ∪ C| = total arrangements, which would mean there are no arrangements where none of the pairs are adjacent. But this contradicts the gap method. Hence, there must be a miscalculation in inclusion-exclusion.

Wait, let me try to calculate |A ∪ B ∪ C| using the gap method. If the gap method gives 144 arrangements where none of the pairs are adjacent, then the inclusion-exclusion should give 720 - 144 = 576. Therefore, let's see:

If |A ∪ B ∪ C| = 720 - 144 = 576, then according to inclusion-exclusion:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720 is wrong. So what is correct? Let me recast.

Wait, no. The correct value of |A ∪ B ∪ C| is 720 - 144 = 576, so according to inclusion-exclusion:

576 = 720 - 144 + |A ∩ B ∩ C|.

Wait, no, the inclusion-exclusion formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, if |A ∪ B ∪ C| = 576, then:

576 = 720 - 144 + |A ∩ B ∩ C|.

Solving for |A ∩ B ∩ C|:

|A ∩ B ∩ C| = 576 - 720 + 144 = 0.

But that would mean |A ∩ B ∩ C| = 0, which is not possible, because there are arrangements where all three are together. Therefore, this is a contradiction.

This suggests that there's a fundamental error in the inclusion-exclusion approach, possibly due to miscalculating the intersections.

Given the time I've spent and the confusion, perhaps I should trust the gap method, since in the n=4 case it worked, and it's a standard method for non-adjacent problems.

So, going back to the gap method: arranging the 4 non-P-R-T people around the table, then placing P, R, T in the gaps.

Number of ways to arrange the 4 people around the table: (4-1)! = 6.

Number of gaps: 4.

Number of ways to choose 3 gaps out of 4: C(4,3) = 4.

Number of ways to arrange P, R, T in those gaps: 3! = 6.

Total: 6 * 4 * 6 = 144.

But let me verify this with another approach. For example, using the inclusion-exclusion but correctly calculating the values.

Let me try to calculate the number of arrangements where no two of P, R, T are adjacent, which is what we want.

Alternatively, using the gap method: it's (number of ways to arrange the others) * (number of ways to place the three people in the gaps).

Number of ways to arrange the other 4 people: (4-1)! = 6.

Number of ways to choose gaps: C(4,3) = 4.

Number of ways to arrange P, R, T in the gaps: 3! = 6.

Total: 6 * 4 * 6 = 144.

However, let me check for n=4, as before. With n=4, the other people is 1, so arranging 1 person: (1-1)! = 1. Number of gaps: 1. Number of ways to choose 3 gaps: C(1,3) = 0. Hence, total arrangements is 0. Which matches the correct result. So, for n=7, it gives 144, which seems plausible.

Alternatively, another way to think is: the number of ways to arrange 7 people around a table is 6! = 720. The number of ways where P, R, T are all separated is equal to (number of ways to arrange the others) * (number of ways to insert P, R, T into the gaps). So, arranging 4 people around the table: (4-1)! = 6. Then, between them, there are 4 gaps. We need to place 3 people into these gaps, one per gap, which is P(4,3) = 4 * 3 * 2 = 24, but wait, no, it's C(4,3) * 3! = 4 * 6 = 24. So total is 6 * 24 = 144. Yes, same as before. So, this seems correct.

Therefore, likely the correct answer is 144. But why does the inclusion-exclusion give conflicting results? Perhaps because of miscalculating |A ∩ B ∩ C|?

Wait, let me try to calculate |A ∩ B ∩ C| correctly. |A ∩ B ∩ C| is the number of arrangements where all three pairs are adjacent, i.e., all three are adjacent. How many such arrangements are there?

Treat P, R, T as a block. Number of entities: 5 (block + A, B, C, D). Number of circular arrangements: (5-1)! = 24. Multiply by internal arrangements of the block: 3! = 6, giving 24 * 6 = 144. But this counts all arrangements where P, R, T are together, regardless of their order. However, when we calculated |A ∩ B|, we had only 2 arrangements for the block, but now it's 6. Why is that?

Because |A ∩ B| is the number of arrangements where P is adjacent to R and P is adjacent to T, which allows the block to be arranged as R-P-T or T-P-R, but not all permutations. Whereas |A ∩ B ∩ C| is the number of arrangements where all three pairs are adjacent, which requires all three to be together in any order, hence 6 internal arrangements.

Therefore, in the inclusion-exclusion formula, if we have |A ∪ B ∪ C| = 720 - 144 + 144 = 720, this suggests that every arrangement is in at least one of A, B, C, which is false. Therefore, there must be an error in the calculation of |A|, |B|, |C|. Let me check |A| again.

If |A| is the number of arrangements where P and R are adjacent, calculated as 240, but let's verify it with fixing P's position. Fix P's position. Then, R can be in 2 seats adjacent to P. The remaining 5 people (including T, A, B, C, D) can be arranged in the remaining 5 seats: 5! = 120. Therefore, total |A| = 2 * 120 = 240. This seems correct. Similarly for |B| and |C|. But if every arrangement has at least one pair, then the number of arrangements where all three are separated would be zero, but the gap method says 144. So there's a contradiction.

Wait, but if I calculate using the gap method, the number of arrangements with no two of P, R, T adjacent is 144, then by complementary counting, the number of arrangements with at least one pair is 720 - 144 = 576. Let's see if that matches the inclusion-exclusion.

If |A ∪ B ∪ C| = 576, then according to inclusion-exclusion:

576 = 720 - 144 + |A ∩ B ∩ C|.

Solving for |A ∩ B ∩ C|:

|A ∩ B ∩ C| = 576 - 720 + 144 = 0.

But this is impossible, since there are arrangements where all three are together. Hence, this suggests that there is a miscalculation in |A ∩ B| or |A ∩ C| or |B ∩ C|.

Wait, let me recalculate |A ∩ B|. If |A ∩ B| is the number of arrangements where both P-R and P-T are adjacent, which means P is between R and T, forming a block of R-P-T or T-P-R. Let's calculate this using fixing P's position.

Fix P's position. Then, R and T must be in the two seats adjacent to P. There are 2 seats: one on the left, one on the right. R and T can be arranged in 2! = 2 ways. The remaining 4 people (A, B, C, D) can be arranged in the remaining 4 seats in 4! = 24 ways. So total is 2 * 24 = 48, as before.

Now, how many times is each arrangement counted in |A|, |B|, |C|? For example, an arrangement where P-R-T are all together is counted in |A| (P-R), |B| (P-T), and |C| (R-T). Similarly, an arrangement where P is adjacent to R and T but not R and T is counted in |A| and |B|.

But the inclusion-exclusion formula should account for this. Let me try to calculate |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

If the correct value of |A ∪ B ∪ C| is 720 - 144 = 576, then:

576 = 720 - 144 + |A ∩ B ∩ C| => |A ∩ B ∩ C| = 576 - 720 + 144 = 0. Which is impossible. Hence, the error must be elsewhere.

Wait, let me calculate |A ∩ B ∩ C| using another method. |A ∩ B ∩ C| is the number of arrangements where P-R, P-T, and R-T are all adjacent, which means P, R, T are all together. As calculated earlier, it is 144. If we substitute this into the inclusion-exclusion formula:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. But this suggests that all arrangements are in |A ∪ B ∪ C), which would mean that there are no arrangements with none of the pairs adjacent, which contradicts the gap method. Hence, there is a contradiction, indicating a fundamental error in one of the methods.

However, given that the gap method works for n=4 and seems logically sound, I'm inclined to trust it. Perhaps the error is in the inclusion-exclusion approach, possibly due to overcounting or undercounting in the intersection terms. Given the time I've spent and the consistency of the gap method, I will go with the result from the gap method: 144.

But wait, let me check another way. Let's think about the total number of ways where no two of P, R, T are adjacent is equal to the total number of seating arrangements minus the number of seating arrangements where at least one pair is adjacent. If the gap method says 144, then the number of arrangements with at least one pair is 720 - 144 = 576. Let me calculate this number using inclusion-exclusion, assuming that |A ∪ B ∪ C| = 576.

If |A ∪ B ∪ C| = 576, then:

576 = 720 - 144 + |A ∩ B ∩ C| => |A ∩ B ∩ C| = 576 - 720 + 144 = 0. Which is impossible, so this suggests that the error is in the calculation of |A ∩ B|, |A ∩ C|, |B ∩ C|. Let me think: if |A ∩ B| is not 48, but something else.

Wait, if |A ∩ B| is the number of arrangements where both P-R and P-T are adjacent, which is the same as the number of arrangements where P is adjacent to both R and T, which is the same as the number of arrangements where P, R, T are all adjacent. Wait, no! If P is adjacent to both R and T, it doesn't necessarily mean that R and T are adjacent. For example, P could be between R and T, but R and T are not adjacent. Wait, no, in a circular table, if P is adjacent to both R and T, then R and T are adjacent to P, but are they adjacent to each other? No, unless the three are all together. Wait, if P is between R and T, then R and T are adjacent through P. Wait, in a circle, if P is between R and T, then R and T are not adjacent. Wait, no. If you have a circular table, and you have R, P, T in a row, then R and T are not adjacent, unless the seating is such that R and T are next to each other. Wait, no, if you have R, P, T in a row, then R is next to P, and P is next to T, but R and T are not next to each other. Therefore, |A ∩ B| is the number of arrangements where P is adjacent to R and P is adjacent to T, but R and T are not necessarily adjacent. Therefore, this is different from |A ∩ B ∩ C|, which requires R and T to be adjacent as well.

Therefore, my earlier calculation of |A ∩ B| as 48 was incorrect. Because when I fixed P's position and said that R and T are in the two seats next to P, that actually forces R and T to be adjacent to P, but not necessarily to each other. Wait, no, if P is fixed, and R is on one side, T on the other, then R and T are separated by P, so they are not adjacent. Therefore, in this case, |A ∩ B| is the number of arrangements where P is adjacent to both R and T, but R and T are not adjacent. Therefore, this is different from the case where all three are adjacent.

Therefore, my earlier calculation of |A ∩ B| as 48 is correct, but |A ∩ B ∩ C| is the number of arrangements where P is adjacent to R, P is adjacent to T, and R is adjacent to T, which is the same as all three being adjacent. So, the number of such arrangements is 144, as calculated.

But then, how does this reconcile with the inclusion-exclusion? Let me try again.

If |A| = 240, |B| = 240, |C| = 240.

|A ∩ B| = 48, |A ∩ C| = 48, |B ∩ C| = 48.

|A ∩ B ∩ C| = 144.

Then, inclusion-exclusion formula gives:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. Which implies that all arrangements have at least one pair, which contradicts the gap method. But if |A ∩ B| is not 48, but something else?

Wait, let me recalculate |A ∩ B| using the fixed P method. If P is fixed, number of arrangements where P is adjacent to R and P is adjacent to T is 2 * 4! = 48, as before. But how many such arrangements are there in total? Let's think: for each arrangement where P is adjacent to R and P is adjacent to T, R and T are not adjacent. And there are also arrangements where all three are adjacent, which are counted in |A ∩ B ∩ C|. So, |A ∩ B| = (arrangements where P is adjacent to R and P is adjacent to T, but R and T are not adjacent) + (arrangements where all three are adjacent).

Wait, but earlier calculation of |A ∩ B| as 48 includes both cases. But if we calculate |A ∩ B| as the number of arrangements where P is adjacent to both R and T, it includes both cases where R and T are adjacent and not adjacent. But how?

Wait, if P is adjacent to R and P is adjacent to T, and the seating is circular, then R and T can either be on opposite sides of P (not adjacent) or if the table is small enough, they could be adjacent. Wait, in a 7-seat table, if P is fixed, and R and T are on either side of P, they are not adjacent. If the block of P, R, T is placed such that R and T are adjacent, then it would require that R and T are next to each other, which would only occur if the block is of size 3, but in that case, P is between R and T, so R and T are not adjacent. Wait, this is confusing.

Wait, if P is between R and T, then R and T are not adjacent. But if the block is R-P-T, then R and T are separated by P. So, in this case, R and T are not adjacent. Therefore, |A ∩ B| is the number of arrangements where P is adjacent to both R and T, and R and T are not adjacent. Therefore, |A ∩ B| = 48, and |A ∩ B ∩ C| is the number of arrangements where all three are adjacent, which is 144. But how does this fit into inclusion-exclusion?

If |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

If |A ∩ B| is the number of arrangements where P is adjacent to both R and T (but R and T not adjacent), and |A ∩ B ∩ C| is the number where all three are adjacent, then the formula should still hold. But if |A ∪ B ∪ C| = 720, it suggests that every arrangement has at least one pair, which is not true. Hence, there must be a miscalculation.

At this point, given the time I've spent and the consistency of the gap method, I will proceed with the result from the gap method, which gives 144. Therefore, the number of ways is 144. However, to ensure, let me check with another approach.

Let me calculate the number of ways using the gap method:

First, arrange the 4 non-P-R-T people around the table: (4-1)! = 6 ways.

Now, there are 4 gaps between them. We need to place 3 people (P, R, T) into these gaps, with no more than one person per gap. The number of ways to choose 3 gaps from 4 is C(4,3) = 4. Then, arrange P, R, T in these gaps: 3! = 6. So total ways: 6 * 4 * 6 = 144.

Yes, this seems correct.

Alternatively, if I think of it as first arranging the 4 people, then placing P, R, T in the gaps, which is a standard non-adjacent arrangement technique.

Therefore, I conclude that the correct answer is 144. The inclusion-exclusion approach must have had an error in the calculation of intersections, possibly due to complex dependencies, and the gap method is more straightforward and verified with smaller cases.

**Final Answer**
The number of ways is \boxed{144}.

To determine the number of ways that **7 people** can sit around a **circular table** such that **no two of the three specific individuals (Pierre, Rosa, and Thomas)** sit next to each other, we approach the problem methodically using the **gap method** — a standard combinatorial technique for avoiding adjacency under circular constraints.

---

### 🧠 **Understanding the Problem:**

- There are **7 people** in total: P, R, T (3 individuals who **cannot sit next to each other**), and A, B, C, D (4 others).
- A **circular table** means that arrangements that are **rotations of each other are considered identical**.
- The **specific condition** is that no two of **P, R, T** can be adjacent.
- We are to count how many **distinct seating arrangements** satisfy this condition.

---

### 📌 **Step 1: Fix the Seating of the 4 People Who Are Not P, R, T**

To simplify the problem, we first consider the **arrangements of the 4 people who are not restricted** — A, B, C, and D.

- Since the table is circular, the number of **distinct arrangements** of these 4 people is:
  $$
  (4 - 1)! = 3! = 6
  $$

---

### 📌 **Step 2: Count the Available Gaps for P, R, T**

Once A, B, C, D are seated around the table, there are exactly **4 gaps** (one between each pair of people), and each seat is adjacent to two others.

- To ensure that **P, R, and T are not adjacent**, we must place **each of them in a different gap**.

- So, we choose **3 of the 4 gaps** to place P, R, and T:
  $$
  \binom{4}{3} = 4
  $$

---

### 📌 **Step 3: Arrange P, R, T in the Chosen Gaps**

We now arrange **P, R, and T** in the selected gaps in all possible orders:
$$
3! = 6
$$

---

### 📌 **Step 4: Multiply All Possibilities**

Now, multiplying the number of arrangements of the unrestricted individuals with the choices and placements of P, R, and T:
$$
3! \text{ (for the 4 people)} \times \binom{4}{3} \text{ (choose gaps)} \times 3! \text{ (arrange P, R, T)}
= 6 \times 4 \times 6 = \boxed{144}
$$

---

### ✅ **Final Answer**:

$$
\boxed{144}
$$

This is the total number of distinct seating arrangements of the 7 people such that no two of Pierre, Rosa, and Thomas sit adjacent to each other around the circular table.

[Assistant]
Explanation:
Okay, so I need to figure out how many ways 7 people can sit around a round table, but with the condition that Pierre, Rosa, and Thomas (let me call them P, R, T for short) can't sit next to each other. Also, rotations of the same arrangement are considered the same, so it's a circular permutation problem. Alright, let me start by recalling some basics about circular permutations.

First, for circular permutations, the number of ways to arrange n people around a round table is (n-1)! because rotations are considered the same. So, if there were no restrictions, the total number of arrangements would be (7-1)! = 6! = 720. But here, we have restrictions: P, R, and T can't sit next to each other. So, I need to subtract the arrangements where at least two of them are sitting together. Wait, but inclusion-exclusion might be necessary here. Hmm.

Alternatively, maybe it's easier to first arrange the other people and then place P, R, T in the gaps? Let me think. That's a common technique for problems where certain people can't be adjacent. Let me try that approach.

So, there are 7 people total. Let's consider the other 4 people (not P, R, T) first. Let me call them A, B, C, D. If we fix the positions of A, B, C, D around the table, then we can place P, R, T in the gaps between them. However, since it's a circular table, the number of gaps between the 4 people would be equal to 4. Wait, but arranging 4 people around a circular table: normally, the number of arrangements is (4-1)! = 6. But hold on, if we fix the positions of A, B, C, D, then the number of arrangements is actually (4-1)! = 6? Wait, no, if we fix one person to account for circularity, then the rest can be arranged linearly. Wait, maybe I need to think more carefully.

Wait, actually, when arranging people around a circular table, fixing one person's position removes the rotational symmetry. So, if I fix one person, say A, then arrange the remaining 3 people around the table. But perhaps I need to consider arranging A, B, C, D first. However, since we need to place P, R, T in the gaps between them, maybe it's better to first seat the 4 non-P-R-T people, then place P, R, T in the gaps.

But in circular permutations, when we arrange the 4 people, the number of ways is (4-1)! = 6. Then, once they are seated, there are 4 gaps between them where we can place the other people. So, if we have 4 gaps, and we need to place 3 people (P, R, T) into these gaps such that no two are in the same gap (since if two are in the same gap, they would be adjacent). Wait, but since the table is circular, arranging the 4 people creates 4 gaps between them. Each gap is between two people.

Therefore, if I want to place P, R, T into these gaps, such that each is in a separate gap, then the number of ways would be: first, arrange the 4 people around the table (which is (4-1)! = 6 ways), then choose 3 gaps out of the 4, and arrange P, R, T in those gaps. So, the number of ways would be 6 * C(4,3) * 3!.

Calculating that: 6 * 4 * 6 = 6 * 24 = 144? Wait, C(4,3) is 4, and then 3! is 6, so 4*6=24, times 6 is 144. So, total number of arrangements would be 144? But wait, the total number of unrestricted arrangements is 720. So, 144 is less than that. Wait, but is this the correct approach?

Wait, but hold on. If we first fix the positions of the 4 people around the table, then arrange the others. However, in circular permutations, when we fix one person, we're already accounting for rotational symmetry. But if we first fix the 4 people, then the number of ways is (4-1)! as before. But then placing P, R, T into gaps. Wait, but another thought: when placing P, R, T into gaps, since the table is circular, once we fix the positions of the 4 people, the gaps are fixed. Therefore, the number of ways to place P, R, T into the gaps without them sitting together is C(4,3)*3!.

But is this correct? Let me think again. If there are 4 gaps, each gap can hold any number of people, but if we want to prevent any two of P, R, T from sitting next to each other, we need each of them in separate gaps. Therefore, yes, we have to choose 3 gaps out of 4, and then arrange P, R, T in those gaps. Each such arrangement would result in P, R, T not sitting next to each other. Therefore, the total number is indeed (number of ways to arrange the 4 people) * (number of ways to choose gaps) * (number of ways to arrange P, R, T in those gaps).

But let me verify this approach with an example. Suppose we have 4 people around the table: A, B, C, D. Let me fix their positions. For example, fix A at a position, then B, C, D can be arranged in 3! = 6 ways. Then there are 4 gaps: between A & B, B & C, C & D, D & A. So, if we choose any 3 of these gaps, say the first, second, and third gaps, and place P, R, T in them. Then, each of P, R, T is separated by at least one person. Therefore, they are not adjacent. Therefore, this approach should work.

But then, the total number is 6 * 4 * 6 = 144. However, let me check if there's another approach. Alternatively, maybe using inclusion-exclusion. Let me try that as well to see if I get the same answer.

Total number of circular arrangements: (7-1)! = 720.

Now, subtract the arrangements where at least two of P, R, T are sitting together. Wait, but inclusion-exclusion can get complicated here. Let me try.

First, compute the number of arrangements where at least one pair of P, R, T are sitting together. Let me denote the three people as P, R, T. Let me define:

A: arrangements where P and R are sitting together.

B: arrangements where P and T are sitting together.

C: arrangements where R and T are sitting together.

We need to compute |A ∪ B ∪ C|, and subtract that from the total to get the number of arrangements where none of the pairs are sitting together.

By inclusion-exclusion:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, first compute |A|. How many circular arrangements have P and R sitting together?

To compute |A|, treat P and R as a single entity. Then, we have 6 entities to arrange around the table: the PR pair, T, A, B, C, D. However, since it's circular, the number of arrangements is (6-1)! = 120. But also, the PR pair can be in two orders: PR or RP. So, |A| = 2 * 5! = 2 * 120 = 240? Wait, 5! is 120, so 2 * 120 = 240? Wait, but hold on: Wait, the formula for circular permutations when we have a pair is (n-1)! for the number of arrangements of the entities, times the internal arrangements. So yes, for each pair, it's 2*(n-2)! ?

Wait, let me think. If we have n people, and we consider two people as a single entity, then the number of circular arrangements is (n-1)! for the entities, but since the two people can be arranged in 2 ways, it's 2*(n-1)!? Wait, but actually, when you have a pair, the number of entities is (n - 1), because we merged two into one. Therefore, the number of circular arrangements would be (n - 2)! * 2? Wait, no:

Wait, the standard formula for circular permutations with some people together is: treat the group as one entity, so total entities = n - 1. Then the number of circular arrangements is (n - 2)! * 2 (for the internal arrangement). Wait, let me check with n=3: if we have 3 people, and want to count arrangements where two are together. Treating them as a pair, then number of circular arrangements is (2-1)! * 2 = 1 * 2 = 2. But in reality, for 3 people around a table, there are (3-1)! = 2 arrangements. However, if we consider two people fixed together, then there's only 1 way to arrange them (since rotations are considered same). Wait, this is confusing.

Wait, maybe I need to think differently. Let me take a concrete example. Suppose n=3 people: A, B, C. How many circular arrangements where A and B are together? Well, in circular arrangements, if we fix A's position, then B can be to the left or right of A, and C is fixed. So, there are 2 arrangements. But according to the formula (n - 2)! * 2: n=3, so (3-2)! * 2 = 1 * 2 = 2. Correct. But if we didn't fix positions, the total number of circular arrangements for 3 people is 2. So, yes, seems like the formula works. So in general, for circular arrangements where a specific pair is together, it's 2*(n - 2)! ?

Wait, but in our case, n=7, so for |A|, the number of arrangements where P and R are together is 2*(7 - 2)! = 2*5! = 240? Wait, but hold on, when we merge P and R into a single entity, we have 6 entities: the PR pair, T, A, B, C, D. Then, the number of circular arrangements is (6 - 1)! = 5! = 120. Multiply by 2 for the internal arrangement of PR, so 240. So yes, that's |A|.

Similarly, |B| and |C| are the same. So, each of |A|, |B|, |C| is 240. Therefore, |A| + |B| + |C| = 3 * 240 = 720.

But wait, that's equal to the total number of circular arrangements? That seems odd. Because 720 is the total number of arrangements. But if we subtract |A ∪ B ∪ C| from total arrangements, which would be 720 - |A ∪ B ∪ C|, but if |A ∪ B ∪ C| is 720, that would imply zero, which can't be. So, clearly, something is wrong here. Wait, but |A| is 240, but if we have three such sets, their total is 720, but they have overlaps. So, we need to subtract the intersections. So, let me continue.

Now, |A ∩ B|: arrangements where both P and R are together, and P and T are together. Wait, if P is with both R and T, that means P is between R and T, so effectively, the three of them form a block. Similarly, |A ∩ B| would be the number of arrangements where P is adjacent to both R and T, meaning that P, R, T are all sitting together in a block. Similarly for other intersections. Let me confirm.

Yes, because if P and R are together, and P and T are together, then R and T must also be adjacent to P, so all three are in a block. Therefore, |A ∩ B| is equal to the number of circular arrangements where P, R, T are all together. Similarly, |A ∩ C| would be the same: arrangements where P and R are together and R and T are together, which again implies all three are together. Similarly |B ∩ C| is same. So each |A ∩ B|, |A ∩ C|, |B ∩ C| is equal to the number of arrangements where P, R, T are all together.

So, let's compute that. How many circular arrangements have P, R, T all together. Treat them as a single block. Then, we have 5 entities: the PRT block, A, B, C, D. Number of circular arrangements is (5-1)! = 4! = 24. Then, the block PRT can be arranged in 3! = 6 ways. Therefore, total |A ∩ B| = 6 * 4! = 6 * 24 = 144. Wait, 4! is 24? Wait, no. Wait, (5-1)! is 4! = 24. Multiply by 3! for arranging the block. So 24 * 6 = 144. But each intersection |A ∩ B|, |A ∩ C|, |B ∩ C| is 144. Therefore, the total |A ∩ B| + |A ∩ C| + |B ∩ C| = 3 * 144 = 432.

Now, the inclusion-exclusion formula says:

|A ∪ B ∪ C| = 720 - 432 + |A ∩ B ∩ C|.

Wait, no. Wait, the formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, we have:

|A ∪ B ∪ C| = 720 - 432 + |A ∩ B ∩ C|.

Wait, no. Wait, original formula:

|A ∪ B ∪ C| = (sum |A|) - (sum |A ∩ B|) + |A ∩ B ∩ C|.

So substituting numbers:

= 720 - 432 + |A ∩ B ∩ C|.

But we need to compute |A ∩ B ∩ C|. This is the number of arrangements where P and R are together, P and T are together, and R and T are together. Which is exactly the same as all three being together. So, |A ∩ B ∩ C| is same as the number of arrangements where all three are together, which is same as before. Wait, but in the previous calculation, when we calculated |A ∩ B|, we considered that as all three together. So, |A ∩ B ∩ C| is also the same, because if all three pairs are together, then all three must be together. Therefore, |A ∩ B ∩ C| is same as the number of arrangements where all three are together, which is 144? Wait, no. Wait, no. Wait, when we calculated |A ∩ B|, we treated the block of PRT as one entity, leading to 4! * 3! = 24 * 6 = 144. However, in reality, if all three are together, then the number of circular arrangements is (number of ways to arrange the block and other people) * (internal arrangements). Wait, but if I have 5 entities: the PRT block, A, B, C, D. Then, circular arrangements: (5 - 1)! * 3! = 4! * 6 = 24 * 6 = 144. So yes, same as before.

But that seems like |A ∩ B ∩ C| is also 144? Wait, but if we have all three pairs together, that's the same as having all three together. So |A ∩ B ∩ C| is 144. Therefore, putting it back into inclusion-exclusion:

|A ∪ B ∪ C| = 720 - 432 + 144 = 720 - 432 is 288; 288 + 144 = 432. Wait, so |A ∪ B ∪ C| is 432? Therefore, the number of arrangements where at least one pair of P, R, T are sitting together is 432. Therefore, the number of arrangements where none of them are sitting together is total arrangements minus that, which is 720 - 432 = 288. Wait, but earlier approach using gaps gave me 144. So which is correct?

This discrepancy is concerning. Which one is wrong?

Let me check with small numbers. Let me take a smaller case where I can compute manually. Let me try with n=4 people: P, R, T, and one more person, say A. Let's say we want to find the number of circular arrangements where P, R, T are not sitting together. Wait, but in n=4, there are only 4 seats. If P, R, T are three people, they can't all be separated because there's only 4 seats. Wait, so for n=4, the number of arrangements where no two of P, R, T are adjacent? Let's see. Since there are only 4 seats, if we seat P, R, T, and A. To have no two of P, R, T adjacent, they need to be placed with at least one person between them. But with 4 seats, and 3 people, it's impossible because if you seat them with one seat between each, you need 3 seats for the people and 1 seat between them, but in a circle, that would require 3*2=6 seats? No, wait. For circular arrangements, the number of seats is equal to the number of people. So with 4 seats, how to place 3 people such that none are adjacent? Wait, in a circular table with 4 seats, each seat has two neighbors. If we place 3 people, then at least two of them must be adjacent because if you have 3 people, they occupy 3 seats, leaving 1 seat. The 1 seat is between two people, so those two people are adjacent. Therefore, it's impossible. Therefore, the answer is zero. Let me check with both methods.

First method: using the gap method. Number of ways to arrange the 1 non-P-R-T person (A) around the table: (1-1)! = 1? Wait, but arranging 1 person around a circular table is 0!? Wait, no. Wait, the formula is (n-1)! for n people. So if we have n=1, the number of circular arrangements is 0!? Wait, actually, for n=1, the number of circular arrangements is 1 (just the person sitting alone). So (1-1)! = 0! = 1. So arranging A around the table: 1 way. Then, the number of gaps is equal to 1. Wait, if there is only one person, there is one gap between them (but since it's a circle, it's just one gap). Wait, but we need to place 3 people into these gaps, but there's only 1 gap. Therefore, we can't place them all into separate gaps. Therefore, the number of ways would be zero. Which matches the actual answer. So the gap method gives zero.

Now, using inclusion-exclusion. Total arrangements: (4-1)! = 6. Number of arrangements where at least one pair of P, R, T are adjacent. Let me compute that. Using inclusion-exclusion.

First, |A|, |B|, |C| each: treating a pair as a block. So for |A|: number of arrangements where P and R are together. So treat them as a block, so we have 3 entities: PR, T, A. Number of circular arrangements: (3-1)! = 2. Multiply by 2 for internal arrangements: 2*2=4. Similarly for |B| and |C|. So total |A| + |B| + |C| = 3*4=12. Then, |A ∩ B|, which is arrangements where P and R are together and P and T are together, which implies all three are together. Number of arrangements: treat PRT as a block, so entities are PRT and A. Number of circular arrangements: (2-1)! = 1. Multiply by internal arrangements: 3! = 6. So total |A ∩ B| = 6. Similarly for |A ∩ C| and |B ∩ C|, so total is 3*6=18. Then |A ∩ B ∩ C| is same as all three together, which is 6. Therefore, inclusion-exclusion formula:

|A ∪ B ∪ C| = 12 - 18 + 6 = 0? Wait, that can't be. Wait:

Wait, formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So substituting numbers:

= 12 - 18 + 6 = 0? That would mean that there are no arrangements where at least one pair is together? Which is impossible, since we know that in all arrangements, at least two of P, R, T must be adjacent. Therefore, there must be a mistake in calculations.

Wait, let me recalculate. For n=4, total arrangements is 6. How many of them have at least one pair of P, R, T adjacent? Let's list them.

Since n=4, total circular permutations is 3! = 6. Let me fix A's position to remove rotational symmetry. Let me list all possible arrangements:

Let's fix A at a position. Then, the remaining 3 people are P, R, T.

Possible arrangements:

1. P, R, T, A

2. P, T, R, A

3. R, P, T, A

4. R, T, P, A

5. T, P, R, A

6. T, R, P, A

But since it's a circle, some of these are rotations. Wait, no, since we fixed A's position, these are all distinct.

Wait, but actually, when we fix A's position, the number of arrangements is 3! = 6. Let me check which of these have at least two of P, R, T adjacent.

Arrangement 1: P, R, T, A. So P and R are adjacent, R and T are adjacent, T and A are adjacent. So yes, multiple adjacents.

Arrangement 2: P, T, R, A. P and T are adjacent, T and R are adjacent, R and A are adjacent. Also multiple adjacents.

Arrangement 3: R, P, T, A. R and P adjacent, P and T adjacent, T and A adjacent.

Arrangement 4: R, T, P, A. R and T adjacent, T and P adjacent, P and A adjacent.

Arrangement 5: T, P, R, A. T and P adjacent, P and R adjacent, R and A adjacent.

Arrangement 6: T, R, P, A. T and R adjacent, R and P adjacent, P and A adjacent.

So all 6 arrangements have at least two of P, R, T adjacent. Therefore, |A ∪ B ∪ C| = 6. But according to inclusion-exclusion earlier, I had 0. Which is wrong. Therefore, there's a mistake in my calculation. Let me check again.

So for n=4, |A|: arrangements where P and R are together. How many?

Treating P and R as a block. So the block PR, T, A. Number of circular arrangements: since we fixed A's position, maybe the formula is different?

Wait, earlier I assumed that treating a pair as a block would give (number of entities - 1)! * internal arrangements. But if we fixed A's position, then the number of arrangements where P and R are together would be different?

Wait, maybe I need to adjust how I compute |A| in the inclusion-exclusion.

Alternatively, perhaps my formula for |A| was wrong. Let me think again.

In the case where we have n people, and we want the number of circular arrangements where a specific pair is together. If we fix one person's position, say A, then the total number of arrangements is (n-1)! So here, for n=4, fixing A, total arrangements is 3! = 6.

To compute |A| (P and R together): How many arrangements have P and R adjacent?

If we fix A's position, then we have three seats left: positions 1, 2, 3 (assuming A is fixed at position 4). Let me think of it as a line with fixed A at one end, but since it's a circle, fixing A's position makes it linear.

Wait, maybe it's easier to think linearly. When we fix A's position, the rest can be arranged linearly with the condition that P and R are adjacent.

Number of ways to arrange P, R, T in the remaining 3 seats with P and R adjacent. Let me think. In linear arrangements, with 3 seats, the number of arrangements where P and R are adjacent is 2 * 2! = 4. Wait:

Number of ways where P and R are adjacent: treat them as a block, which can be PR or RP. Then, the block and T can be arranged in 2 positions: block and T, or T and block. So 2 * 2 = 4. Then, since we fixed A's position, the total is 4. Therefore, |A| is 4. Similarly, |B| and |C| are each 4. Therefore, |A| + |B| + |C| = 12. However, since we fixed A's position, the total number of arrangements is 6. Therefore, clearly, my previous calculation was wrong in assuming that |A| is 2*(n - 2)! etc. Because that would have given |A| as 2*(4 - 2)! = 2*2 = 4, which matches here. Wait, but then why did my previous inclusion-exclusion give the wrong result?

Let me recast the inclusion-exclusion for n=4. So:

Total arrangements: 6.

|A| = |B| = |C| = 4.

Sum |A| + |B| + |C| = 12.

Now, |A ∩ B|: arrangements where P and R are together AND P and T are together. Which implies all three are together. How many such arrangements?

If P, R, T are together, then they form a block. With A fixed, we have the block and A. Since it's a circular table with A fixed, the block can be placed in only one position relative to A? Wait, no. Wait, if we fix A's position, then the block can be placed in positions 1-2-3, but since it's a circle, fixing A's position makes it linear? Wait, maybe not. Let me think again. If A is fixed at position 4, the other three positions are 1, 2, 3. If we have a block of P, R, T occupying positions 1, 2, 3 in some order, then there's only one block. But since the block can be arranged in 3! ways, but considering the circular table, is that correct? Wait, no, because A is fixed. So the block can be arranged in 3! ways, but since we fixed A's position, the number of arrangements where P, R, T are together is 3! = 6. But that can't be, since total arrangements is 6. But in reality, all arrangements have P, R, T together? Which is not true. Wait, for example, if we have arrangement P, T, R, A (which is arrangement 2), then P, T, R are in positions 1, 2, 3. But if we have another arrangement, say, P, T, A, R? Wait, no, since A is fixed at position 4, the positions are fixed. So the three seats are positions 1, 2, 3. So any arrangement of P, R, T in these positions would have them all together. Wait, but in that case, if you fix A at position 4, then the three seats are adjacent to each other (since it's a circle). So positions 1, 2, 3 are adjacent in a circle to position 4 (A). Therefore, positions 1, 2, 3 are adjacent in a circle to each other? No. Wait, in a circular table with 4 seats, seat 1 is adjacent to seat 2 and 4. Seat 2 is adjacent to 1, 3, and 4? No, seat 2 is adjacent to seat 1 and seat 3. Seat 3 is adjacent to seat 2 and seat 4. Seat 4 is adjacent to seat 3 and seat 1. So if A is fixed at seat 4, then seats 1, 2, 3 are arranged linearly around the table, with seat 1 adjacent to seat 4 and seat 2, seat 2 adjacent to seat 1 and seat 3, seat 3 adjacent to seat 2 and seat 4. Therefore, seats 1, 2, 3 are not all adjacent to each other. Seat 1 is only adjacent to 2 and 4. Seat 3 is only adjacent to 2 and 4. So if we place P, R, T in seats 1, 2, 3, they are not all adjacent to each other; only seat 1 and 2, and seat 2 and 3 are adjacent. Therefore, in this case, if we have P, R, T in seats 1, 2, 3, then P and R may be adjacent, R and T may be adjacent, but P and T are not adjacent unless they are in adjacent seats. Wait, but in the example where seats 1, 2, 3 have P, T, R: seat 1 is P, seat 2 is T, seat 3 is R. Then P and T are adjacent (seats 1 and 2), T and R are adjacent (seats 2 and 3), but P and R are not adjacent. Therefore, in this arrangement, only two pairs are adjacent. Therefore, if we have all three of P, R, T in seats 1, 2, 3, they are not all adjacent to each other, but two of them are.

Therefore, in this case, the number of arrangements where all three are adjacent would be zero? Wait, but there is no way to have all three adjacent, because to have all three adjacent, they would need to be in seats 1, 2, 3 with each adjacent to the next, but seat 1 and seat 3 are not adjacent. So in a circle of 4 seats, you cannot have three people all adjacent. Therefore, |A ∩ B| would be zero? Wait, but earlier calculation said that treating them as a block gives 3! = 6, but that's incorrect. So my mistake was in the assumption that treating them as a block would give valid arrangements. Actually, in this case, because of the circular table, when we fix A's position, you cannot have all three P, R, T together. Therefore, my formula was wrong.

This shows that my previous approach for inclusion-exclusion was incorrect. So how do I correctly calculate |A ∩ B|?

This suggests that my initial method for calculating |A ∩ B| was wrong. Therefore, going back to the original problem with n=7, my inclusion-exclusion approach might have been flawed. Therefore, maybe the gap method was correct?

Wait, going back to the original problem. If using the gap method, with 4 people seated, creating 4 gaps, placing 3 people into separate gaps, giving 6 * 4 * 6 = 144. But according to inclusion-exclusion with n=4, this method gives zero, which is correct. But with the inclusion-exclusion approach, I had conflicting results. Therefore, maybe the gap method is correct and the inclusion-exclusion needs to be carefully calculated.

Alternatively, maybe the problem arises because when we calculated |A ∩ B| in the original problem, I treated it as arrangements where all three are together, but in reality, there might be overcounting? Or undercounting?

Alternatively, perhaps the correct way is to use the gap method for the original problem, which gave 144, and then check with another approach.

But let's see. If for n=7, the answer is 144, but when using inclusion-exclusion with n=4, we saw that the gap method gives the correct answer. However, the inclusion-exclusion approach needs to be correctly calculated.

Alternatively, maybe I made a mistake in calculating |A ∩ B| in the original problem.

Let me try again for the original problem (n=7). Let me recast the inclusion-exclusion correctly.

Total number of circular arrangements: 6! = 720.

Now, |A|: arrangements where P and R are adjacent. Treat them as a block. So, number of entities: 6 (block, T, A, B, C, D). Number of circular arrangements: (6 - 1)! = 120. Multiply by 2 (for PR or RP): 240. So |A| = 240. Similarly |B| and |C| = 240 each. So sum is 720.

Now, |A ∩ B|: arrangements where P and R are adjacent AND P and T are adjacent. Which means that P is adjacent to both R and T, so P, R, T are all adjacent in a block. Therefore, treat P, R, T as a single block. Then, number of entities is 5: block, A, B, C, D. Number of circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of the block. How many ways to arrange P, R, T in the block? Since P is adjacent to both R and T, the block can be arranged as P-R-T, P-T-R, R-P-T, T-P-R, R-T-P, T-R-P. Wait, but if P must be adjacent to both R and T, then the three people can be arranged in the block in 2 * 2 = 4 ways? Wait, no. Let me think. If we have a block of P, R, T where P is adjacent to both R and T, then the block must be a sequence where P is between R and T. So, the block can be R-P-T or T-P-R. Therefore, two possibilities. Alternatively, if the block is a circle? Wait, no. The block is linear because it's part of a circular table. Wait, no, the block is part of the circular table. Wait, when we treat P, R, T as a block, the block itself is linear? Or can it be circular? Wait, no, because we are arranging them in a circular table. However, the block is just a linear block. Wait, for the purposes of counting, when we have a block of P, R, T, the internal arrangements are 3! = 6, since they can be in any order. However, in our case, the condition is that P is adjacent to both R and T. Therefore, the block must include P adjacent to both R and T, which would require that in the block, P is in the middle. So, for example, R-P-T or T-P-R. Therefore, only 2 possible arrangements for the block. Therefore, internal arrangements are 2, not 6. Therefore, |A ∩ B| is (number of circular arrangements of entities) * internal arrangements = 4! * 2 = 24 * 2 = 48? Wait, hold on. Wait, number of entities is 5 (block, A, B, C, D). Number of circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of the block: 2. Therefore, |A ∩ B| = 24 * 2 = 48. Similarly for |A ∩ C| and |B ∩ C|.

Wait, previously I had said |A ∩ B| was 144, which was incorrect because I forgot that the internal arrangements are not 6 but only 2. So earlier mistake was in assuming that the entire block of P, R, T can be arranged in 3! ways, but actually, when we are calculating |A ∩ B|, which is arrangements where P is adjacent to both R and T (i.e., P is between R and T), so the internal arrangements of the block are only 2, not 6. Therefore, the correct value of |A ∩ B| is 24 * 2 = 48. Similarly for the other intersections. Therefore, the total |A ∩ B| + |A ∩ C| + |B ∩ C| = 3 * 48 = 144.

Then, |A ∩ B ∩ C|: arrangements where P is adjacent to R, P is adjacent to T, and R is adjacent to T. So all three are adjacent to each other, meaning they form a block. Now, how many internal arrangements? Since all three are adjacent to each other, they can be arranged in any order around the block. However, since it's a circular table, the block itself is linear? Or does it matter?

Wait, when we have a block of P, R, T where all three are adjacent to each other, the block can be arranged in 3! = 6 ways. So, number of circular arrangements is (number of entities: 5, the block, A, B, C, D) circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of block: 6. Therefore, |A ∩ B ∩ C| = 24 * 6 = 144.

Wait, but earlier when calculating |A ∩ B|, we had only 2 arrangements for the block. But for |A ∩ B ∩ C|, we have all three adjacent, so the block can be arranged in 3! ways. So, that seems okay.

Therefore, going back to inclusion-exclusion formula:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| = 720 - 144 + 144 = 720? Wait, no:

Wait, original values:

|A| + |B| + |C| = 720

|A ∩ B| + |A ∩ C| + |B ∩ C| = 3 * 48 = 144

|A ∩ B ∩ C| = 144

Therefore,

|A ∪ B ∪ C| = 720 - 144 + 144 = 720? Which can't be, because that would imply that all arrangements have at least one pair, which is not true. For example, there might be arrangements where P, R, T are all separated.

But according to the gap method, there are 144 such arrangements. Therefore, there's a contradiction here. Therefore, clearly, there's still a mistake in my calculations.

Wait, let's recast the inclusion-exclusion with corrected numbers.

Let me compute again:

Total arrangements: 720

|A| = |B| = |C| = 240, so sum is 720.

Each |A ∩ B| = |A ∩ C| = |B ∩ C| = ?

Earlier, we said that when we have both P-R and P-T adjacent, which implies P is between R and T, so the block is R-P-T or T-P-R, so internal arrangements: 2. Hence, |A ∩ B| = (number of circular arrangements with block) * internal arrangements. The number of circular arrangements with the block is (number of entities - 1)! where entities are 5 (block + A, B, C, D). Hence (5 - 1)! = 24, multiplied by 2 gives 48. Therefore, each |A ∩ B| = 48, so total for all intersections is 3 * 48 = 144.

Then, |A ∩ B ∩ C|: arrangements where all three pairs are adjacent, meaning the three are in a block, which can be arranged in 3! = 6 ways. Number of circular arrangements with the block: (5 - 1)! = 24, multiplied by 6 gives 24 * 6 = 144. Therefore, the inclusion-exclusion formula:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. Wait, that can't be. Therefore, the inclusion-exclusion formula says that |A ∪ B ∪ C| = 720 - 144 + 144 = 720? Which suggests that all arrangements have at least one pair of P, R, T adjacent, which contradicts the gap method which gave 144 arrangements where none are adjacent. Therefore, there is a fundamental error here.

Wait, let's think again. Where is the error?

Perhaps in calculating |A ∩ B|. If |A ∩ B| is arrangements where both P-R and P-T are adjacent, which implies that P is adjacent to both R and T, so they form a block of size 3. However, in such case, the block can be arranged in 2 ways: R-P-T or T-P-R. But when we calculate the number of arrangements with the block, we have (number of entities) = 5 (the block, A, B, C, D). Number of circular arrangements: (5 - 1)! = 24. Multiply by internal arrangements of the block: 2. So 24 * 2 = 48. However, is that correct?

Wait, but if we have a block of 3 people, then when we consider circular arrangements, is the block treated as a single entity? Yes. So, the number of circular arrangements for the block and the other 4 people is (5 - 1)! = 24. Then multiply by the internal arrangements of the block. However, if the block is linear, then internal arrangements are 2. If the block is circular, it would be different. Wait, no. When we treat the block as a single entity in the circular arrangement, the block itself is linear, because it's part of the circular table. So the internal arrangements are linear. Therefore, 2 possibilities, as before.

However, if we consider that the block could be arranged in more ways, perhaps?

Wait, suppose the block is R-P-T. Then, if we rotate the entire table, does that count as the same arrangement? No, because we already accounted for circular permutations by fixing the positions. Wait, no, the formula (number of entities - 1)! accounts for rotational symmetry. Therefore, the calculation seems correct.

But then, if |A ∩ B| = 48, and similarly for others, and |A ∩ B ∩ C| = 144, then:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. Which implies that all arrangements have at least one pair, which conflicts with the gap method. Therefore, there must be another error.

Wait, what is |A ∩ B ∩ C|? It is arrangements where all three are adjacent, which we calculated as 144. However, if we calculate the number of arrangements where all three are adjacent, it's 24 * 6 = 144, as previously. But how many total arrangements are there where all three are adjacent? Let me calculate it via gap method. Wait, if all three are adjacent, it's different from the gap method. Wait, but if we use the inclusion-exclusion, and if the result is that all arrangements have at least one pair, then there are zero arrangements where none are adjacent, which contradicts the gap method.

Alternatively, maybe my calculation of |A ∩ B| is wrong. Wait, let me think of another way: how many arrangements have P and R adjacent, and P and T adjacent. Let me think of it as fixing P's position. Since it's a circular table, fix P's position to remove rotational symmetry. Then, the number of arrangements where P is adjacent to both R and T. If P is fixed, then R and T must be in the two seats adjacent to P. There are 2 seats next to P. So R and T must occupy these two seats, which can be arranged in 2! = 2 ways. Then, the other 4 people (A, B, C, D) can be arranged in the remaining 4 seats in 4! = 24 ways. Therefore, total number of arrangements where P is adjacent to both R and T is 2 * 24 = 48. Which matches the previous calculation of |A ∩ B| = 48. Therefore, this seems correct.

Therefore, with this, |A ∩ B| = 48, as calculated. Then, how does the inclusion-exclusion formula proceed.

With |A ∪ B ∪ C| = 720 - 144 + 144 = 720. Which would imply that all arrangements have at least one pair, which can't be. Therefore, there must be an error in the inclusion-exclusion formula.

Wait, maybe the formula is different. Let me recall inclusion-exclusion formula:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, substituting numbers:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. But that suggests that |A ∪ B ∪ C| = 720. But the total number of arrangements is 720, which would mean that every arrangement has at least one pair, but that's not true. So, where is the mistake?

Wait, no! Wait, the formula is |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|. So substituting:

= 720 - 144 + 144 = 720? No, wait:

It's 720 - 144 + 144 = 720? Wait, that would be 720 - 144 is 576, plus 144 is 720. Yes. So this suggests that |A ∪ B ∪ C| = 720, which would mean that every arrangement is counted in at least one of A, B, C. But this contradicts our gap method, which suggests that there are arrangements where none of the pairs are adjacent.

Therefore, there must be an error in the calculation of |A|, |B|, |C|. Let me check.

Earlier, I calculated |A| as 240. Let me verify this with the fixed P method. Fix P's position. Number of arrangements where P and R are adjacent: P is fixed, R can be on either side of P. So 2 choices for R. Then, the remaining 5 people (including T, A, B, C, D) can be arranged in the remaining 5 seats: 5! = 120. Therefore, total number is 2 * 120 = 240. This matches the previous calculation of |A| = 240.

Similarly for |B| and |C|. Therefore, total |A| + |B| + |C| = 720.

But if |A ∪ B ∪ C| = 720, that would mean every arrangement is in at least one of A, B, C. But that is not true. For example, consider an arrangement where P, R, T are all separated. How is this possible?

Wait, in a 7-seat table, if you have 3 people, it's possible to seat them such that none are adjacent. For example, seat them with at least one person between each pair. For example, if we seat them with two seats between each of them. Let's see, with 7 seats, placing 3 people with at least one seat between them. The minimum number of seats needed is 3 + 2 = 5 (each person needs at least one seat between them). Since 7 >= 5, it is possible. Therefore, there must be arrangements where none of the pairs are adjacent. Hence, the inclusion-exclusion must have an error.

Where is the error? Let me think about the calculation of |A ∩ B ∩ C|. Earlier, I said it's 144, but if every arrangement is in |A ∪ B ∪ C|, then |A ∩ B ∩ C| would be the number of arrangements where all three pairs are adjacent, which is the same as all three being adjacent. But how many are those?

Wait, earlier calculation said 144. Let me verify with fixing P's position. Fix P's position. Then, the number of arrangements where all three are adjacent: P, R, T are together. So, the block can be placed in the table. Since P is fixed, the block can be to the left or right of P. Wait, no. If P is fixed, the block of P, R, T can be placed in two directions: P-R-T or P-T-R. But since the table is circular, fixing P's position, the block can extend clockwise or counterclockwise. Wait, but since P is fixed, the block can be placed in the two adjacent seats. Wait, if P is fixed, then to have P, R, T all together, R and T must be in the two neighboring seats of P, but that would only allow the block to be P-R-T or P-T-R, but that would only cover the immediate neighbors of P. However, in a 7-seat table, there are more seats. Wait, actually, no. If we want all three P, R, T to be together, the block can be placed anywhere, not just adjacent to P. So if P is fixed, the block can be placed in different positions.

Wait, this seems complex. Let me think differently. Number of circular arrangements where P, R, T are all adjacent: treat them as a block, so 5 entities. Number of circular arrangements: (5 - 1)! = 24. Multiply by 3! for internal arrangements: 24 * 6 = 144. But if I fix P's position, how many such arrangements are there?

If P is fixed, the block can start at P's position. The block can be P-R-T or P-T-R, extending to the right or left. But since it's a circle, fixing P, the block can be placed in different ways. Wait, fixing P's position, the block can be placed in different positions. Let me think.

Wait, with P fixed, the block of P, R, T can be placed in the table. The length of the block is 3 seats. In a circular table with P fixed, the number of ways to place a block of 3 seats containing P is 7 - 3 = 4? Wait, no. For a circular table, the number of ways to place a block of k seats is n, but since rotations are accounted for by fixing P's position. Wait, this is getting too tangled.

Alternatively, let me use the formula: number of arrangements where all three are together is 144. If total arrangements is 720, and |A ∪ B ∪ C| is 720, then that would mean that all arrangements have at least one pair, which contradicts the gap method. Therefore, there must be a mistake in the inclusion-exclusion calculation.

Wait, perhaps the error is in the calculation of |A ∩ B|. Wait, earlier, when I fixed P's position and calculated |A ∩ B| as 48, but if I use the formula for |A ∩ B| as 2 * (n - 2)! = 2 * 5! = 240? No, that doesn't make sense.

Wait, let me think about the inclusion-exclusion formula again. If |A ∪ B ∪ C| = 720, and the total number of arrangements is 720, then |A ∪ B ∪ C| = total arrangements, which would mean there are no arrangements where none of the pairs are adjacent. But this contradicts the gap method. Hence, there must be a miscalculation in inclusion-exclusion.

Wait, let me try to calculate |A ∪ B ∪ C| using the gap method. If the gap method gives 144 arrangements where none of the pairs are adjacent, then the inclusion-exclusion should give 720 - 144 = 576. Therefore, let's see:

If |A ∪ B ∪ C| = 720 - 144 = 576, then according to inclusion-exclusion:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720 is wrong. So what is correct? Let me recast.

Wait, no. The correct value of |A ∪ B ∪ C| is 720 - 144 = 576, so according to inclusion-exclusion:

576 = 720 - 144 + |A ∩ B ∩ C|.

Wait, no, the inclusion-exclusion formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

So, if |A ∪ B ∪ C| = 576, then:

576 = 720 - 144 + |A ∩ B ∩ C|.

Solving for |A ∩ B ∩ C|:

|A ∩ B ∩ C| = 576 - 720 + 144 = 0.

But that would mean |A ∩ B ∩ C| = 0, which is not possible, because there are arrangements where all three are together. Therefore, this is a contradiction.

This suggests that there's a fundamental error in the inclusion-exclusion approach, possibly due to miscalculating the intersections.

Given the time I've spent and the confusion, perhaps I should trust the gap method, since in the n=4 case it worked, and it's a standard method for non-adjacent problems.

So, going back to the gap method: arranging the 4 non-P-R-T people around the table, then placing P, R, T in the gaps.

Number of ways to arrange the 4 people around the table: (4-1)! = 6.

Number of gaps: 4.

Number of ways to choose 3 gaps out of 4: C(4,3) = 4.

Number of ways to arrange P, R, T in those gaps: 3! = 6.

Total: 6 * 4 * 6 = 144.

But let me verify this with another approach. For example, using the inclusion-exclusion but correctly calculating the values.

Let me try to calculate the number of arrangements where no two of P, R, T are adjacent, which is what we want.

Alternatively, using the gap method: it's (number of ways to arrange the others) * (number of ways to place the three people in the gaps).

Number of ways to arrange the other 4 people: (4-1)! = 6.

Number of ways to choose gaps: C(4,3) = 4.

Number of ways to arrange P, R, T in the gaps: 3! = 6.

Total: 6 * 4 * 6 = 144.

However, let me check for n=4, as before. With n=4, the other people is 1, so arranging 1 person: (1-1)! = 1. Number of gaps: 1. Number of ways to choose 3 gaps: C(1,3) = 0. Hence, total arrangements is 0. Which matches the correct result. So, for n=7, it gives 144, which seems plausible.

Alternatively, another way to think is: the number of ways to arrange 7 people around a table is 6! = 720. The number of ways where P, R, T are all separated is equal to (number of ways to arrange the others) * (number of ways to insert P, R, T into the gaps). So, arranging 4 people around the table: (4-1)! = 6. Then, between them, there are 4 gaps. We need to place 3 people into these gaps, one per gap, which is P(4,3) = 4 * 3 * 2 = 24, but wait, no, it's C(4,3) * 3! = 4 * 6 = 24. So total is 6 * 24 = 144. Yes, same as before. So, this seems correct.

Therefore, likely the correct answer is 144. But why does the inclusion-exclusion give conflicting results? Perhaps because of miscalculating |A ∩ B ∩ C|?

Wait, let me try to calculate |A ∩ B ∩ C| correctly. |A ∩ B ∩ C| is the number of arrangements where all three pairs are adjacent, i.e., all three are adjacent. How many such arrangements are there?

Treat P, R, T as a block. Number of entities: 5 (block + A, B, C, D). Number of circular arrangements: (5-1)! = 24. Multiply by internal arrangements of the block: 3! = 6, giving 24 * 6 = 144. But this counts all arrangements where P, R, T are together, regardless of their order. However, when we calculated |A ∩ B|, we had only 2 arrangements for the block, but now it's 6. Why is that?

Because |A ∩ B| is the number of arrangements where P is adjacent to R and P is adjacent to T, which allows the block to be arranged as R-P-T or T-P-R, but not all permutations. Whereas |A ∩ B ∩ C| is the number of arrangements where all three pairs are adjacent, which requires all three to be together in any order, hence 6 internal arrangements.

Therefore, in the inclusion-exclusion formula, if we have |A ∪ B ∪ C| = 720 - 144 + 144 = 720, this suggests that every arrangement is in at least one of A, B, C, which is false. Therefore, there must be an error in the calculation of |A|, |B|, |C|. Let me check |A| again.

If |A| is the number of arrangements where P and R are adjacent, calculated as 240, but let's verify it with fixing P's position. Fix P's position. Then, R can be in 2 seats adjacent to P. The remaining 5 people (including T, A, B, C, D) can be arranged in the remaining 5 seats: 5! = 120. Therefore, total |A| = 2 * 120 = 240. This seems correct. Similarly for |B| and |C|. But if every arrangement has at least one pair, then the number of arrangements where all three are separated would be zero, but the gap method says 144. So there's a contradiction.

Wait, but if I calculate using the gap method, the number of arrangements with no two of P, R, T adjacent is 144, then by complementary counting, the number of arrangements with at least one pair is 720 - 144 = 576. Let's see if that matches the inclusion-exclusion.

If |A ∪ B ∪ C| = 576, then according to inclusion-exclusion:

576 = 720 - 144 + |A ∩ B ∩ C|.

Solving for |A ∩ B ∩ C|:

|A ∩ B ∩ C| = 576 - 720 + 144 = 0.

But this is impossible, since there are arrangements where all three are together. Hence, this suggests that there is a miscalculation in |A ∩ B| or |A ∩ C| or |B ∩ C|.

Wait, let me recalculate |A ∩ B|. If |A ∩ B| is the number of arrangements where both P-R and P-T are adjacent, which means P is between R and T, forming a block of R-P-T or T-P-R. Let's calculate this using fixing P's position.

Fix P's position. Then, R and T must be in the two seats adjacent to P. There are 2 seats: one on the left, one on the right. R and T can be arranged in 2! = 2 ways. The remaining 4 people (A, B, C, D) can be arranged in the remaining 4 seats in 4! = 24 ways. So total is 2 * 24 = 48, as before.

Now, how many times is each arrangement counted in |A|, |B|, |C|? For example, an arrangement where P-R-T are all together is counted in |A| (P-R), |B| (P-T), and |C| (R-T). Similarly, an arrangement where P is adjacent to R and T but not R and T is counted in |A| and |B|.

But the inclusion-exclusion formula should account for this. Let me try to calculate |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

If the correct value of |A ∪ B ∪ C| is 720 - 144 = 576, then:

576 = 720 - 144 + |A ∩ B ∩ C| => |A ∩ B ∩ C| = 576 - 720 + 144 = 0. Which is impossible. Hence, the error must be elsewhere.

Wait, let me calculate |A ∩ B ∩ C| using another method. |A ∩ B ∩ C| is the number of arrangements where P-R, P-T, and R-T are all adjacent, which means P, R, T are all together. As calculated earlier, it is 144. If we substitute this into the inclusion-exclusion formula:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. But this suggests that all arrangements are in |A ∪ B ∪ C), which would mean that there are no arrangements with none of the pairs adjacent, which contradicts the gap method. Hence, there is a contradiction, indicating a fundamental error in one of the methods.

However, given that the gap method works for n=4 and seems logically sound, I'm inclined to trust it. Perhaps the error is in the inclusion-exclusion approach, possibly due to overcounting or undercounting in the intersection terms. Given the time I've spent and the consistency of the gap method, I will go with the result from the gap method: 144.

But wait, let me check another way. Let's think about the total number of ways where no two of P, R, T are adjacent is equal to the total number of seating arrangements minus the number of seating arrangements where at least one pair is adjacent. If the gap method says 144, then the number of arrangements with at least one pair is 720 - 144 = 576. Let me calculate this number using inclusion-exclusion, assuming that |A ∪ B ∪ C| = 576.

If |A ∪ B ∪ C| = 576, then:

576 = 720 - 144 + |A ∩ B ∩ C| => |A ∩ B ∩ C| = 576 - 720 + 144 = 0. Which is impossible, so this suggests that the error is in the calculation of |A ∩ B|, |A ∩ C|, |B ∩ C|. Let me think: if |A ∩ B| is not 48, but something else.

Wait, if |A ∩ B| is the number of arrangements where both P-R and P-T are adjacent, which is the same as the number of arrangements where P is adjacent to both R and T, which is the same as the number of arrangements where P, R, T are all adjacent. Wait, no! If P is adjacent to both R and T, it doesn't necessarily mean that R and T are adjacent. For example, P could be between R and T, but R and T are not adjacent. Wait, no, in a circular table, if P is adjacent to both R and T, then R and T are adjacent to P, but are they adjacent to each other? No, unless the three are all together. Wait, if P is between R and T, then R and T are adjacent through P. Wait, in a circle, if P is between R and T, then R and T are not adjacent. Wait, no. If you have a circular table, and you have R, P, T in a row, then R and T are not adjacent, unless the seating is such that R and T are next to each other. Wait, no, if you have R, P, T in a row, then R is next to P, and P is next to T, but R and T are not next to each other. Therefore, |A ∩ B| is the number of arrangements where P is adjacent to R and P is adjacent to T, but R and T are not necessarily adjacent. Therefore, this is different from |A ∩ B ∩ C|, which requires R and T to be adjacent as well.

Therefore, my earlier calculation of |A ∩ B| as 48 was incorrect. Because when I fixed P's position and said that R and T are in the two seats next to P, that actually forces R and T to be adjacent to P, but not necessarily to each other. Wait, no, if P is fixed, and R is on one side, T on the other, then R and T are separated by P, so they are not adjacent. Therefore, in this case, |A ∩ B| is the number of arrangements where P is adjacent to both R and T, but R and T are not adjacent. Therefore, this is different from the case where all three are adjacent.

Therefore, my earlier calculation of |A ∩ B| as 48 is correct, but |A ∩ B ∩ C| is the number of arrangements where P is adjacent to R, P is adjacent to T, and R is adjacent to T, which is the same as all three being adjacent. So, the number of such arrangements is 144, as calculated.

But then, how does this reconcile with the inclusion-exclusion? Let me try again.

If |A| = 240, |B| = 240, |C| = 240.

|A ∩ B| = 48, |A ∩ C| = 48, |B ∩ C| = 48.

|A ∩ B ∩ C| = 144.

Then, inclusion-exclusion formula gives:

|A ∪ B ∪ C| = 720 - 144 + 144 = 720. Which implies that all arrangements have at least one pair, which contradicts the gap method. But if |A ∩ B| is not 48, but something else?

Wait, let me recalculate |A ∩ B| using the fixed P method. If P is fixed, number of arrangements where P is adjacent to R and P is adjacent to T is 2 * 4! = 48, as before. But how many such arrangements are there in total? Let's think: for each arrangement where P is adjacent to R and P is adjacent to T, R and T are not adjacent. And there are also arrangements where all three are adjacent, which are counted in |A ∩ B ∩ C|. So, |A ∩ B| = (arrangements where P is adjacent to R and P is adjacent to T, but R and T are not adjacent) + (arrangements where all three are adjacent).

Wait, but earlier calculation of |A ∩ B| as 48 includes both cases. But if we calculate |A ∩ B| as the number of arrangements where P is adjacent to both R and T, it includes both cases where R and T are adjacent and not adjacent. But how?

Wait, if P is adjacent to R and P is adjacent to T, and the seating is circular, then R and T can either be on opposite sides of P (not adjacent) or if the table is small enough, they could be adjacent. Wait, in a 7-seat table, if P is fixed, and R and T are on either side of P, they are not adjacent. If the block of P, R, T is placed such that R and T are adjacent, then it would require that R and T are next to each other, which would only occur if the block is of size 3, but in that case, P is between R and T, so R and T are not adjacent. Wait, this is confusing.

Wait, if P is between R and T, then R and T are not adjacent. But if the block is R-P-T, then R and T are separated by P. So, in this case, R and T are not adjacent. Therefore, |A ∩ B| is the number of arrangements where P is adjacent to both R and T, and R and T are not adjacent. Therefore, |A ∩ B| = 48, and |A ∩ B ∩ C| is the number of arrangements where all three are adjacent, which is 144. But how does this fit into inclusion-exclusion?

If |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

If |A ∩ B| is the number of arrangements where P is adjacent to both R and T (but R and T not adjacent), and |A ∩ B ∩ C| is the number where all three are adjacent, then the formula should still hold. But if |A ∪ B ∪ C| = 720, it suggests that every arrangement has at least one pair, which is not true. Hence, there must be a miscalculation.

At this point, given the time I've spent and the consistency of the gap method, I will proceed with the result from the gap method, which gives 144. Therefore, the number of ways is 144. However, to ensure, let me check with another approach.

Let me calculate the number of ways using the gap method:

First, arrange the 4 non-P-R-T people around the table: (4-1)! = 6 ways.

Now, there are 4 gaps between them. We need to place 3 people (P, R, T) into these gaps, with no more than one person per gap. The number of ways to choose 3 gaps from 4 is C(4,3) = 4. Then, arrange P, R, T in these gaps: 3! = 6. So total ways: 6 * 4 * 6 = 144.

Yes, this seems correct.

Alternatively, if I think of it as first arranging the 4 people, then placing P, R, T in the gaps, which is a standard non-adjacent arrangement technique.

Therefore, I conclude that the correct answer is 144. The inclusion-exclusion approach must have had an error in the calculation of intersections, possibly due to complex dependencies, and the gap method is more straightforward and verified with smaller cases.

**Final Answer**
The number of ways is \boxed{144}.

To determine the number of ways that **7 people** can sit around a **circular table** such that **no two of the three specific individuals (Pierre, Rosa, and Thomas)** sit next to each other, we approach the problem methodically using the **gap method** — a standard combinatorial technique for avoiding adjacency under circular constraints.

---

### 🧠 **Understanding the Problem:**

- There are **7 people** in total: P, R, T (3 individuals who **cannot sit next to each other**), and A, B, C, D (4 others).
- A **circular table** means that arrangements that are **rotations of each other are considered identical**.
- The **specific condition** is that no two of **P, R, T** can be adjacent.
- We are to count how many **distinct seating arrangements** satisfy this condition.

---

### 📌 **Step 1: Fix the Seating of the 4 People Who Are Not P, R, T**

To simplify the problem, we first consider the **arrangements of the 4 people who are not restricted** — A, B, C, and D.

- Since the table is circular, the number of **distinct arrangements** of these 4 people is:
  $$
  (4 - 1)! = 3! = 6
  $$

---

### 📌 **Step 2: Count the Available Gaps for P, R, T**

Once A, B, C, D are seated around the table, there are exactly **4 gaps** (one between each pair of people), and each seat is adjacent to two others.

- To ensure that **P, R, and T are not adjacent**, we must place **each of them in a different gap**.

- So, we choose **3 of the 4 gaps** to place P, R, and T:
  $$
  \binom{4}{3} = 4
  $$

---

### 📌 **Step 3: Arrange P, R, T in the Chosen Gaps**

We now arrange **P, R, and T** in the selected gaps in all possible orders:
$$
3! = 6
$$

---

### 📌 **Step 4: Multiply All Possibilities**

Now, multiplying the number of arrangements of the unrestricted individuals with the choices and placements of P, R, and T:
$$
3! \text{ (for the 4 people)} \times \binom{4}{3} \text{ (choose gaps)} \times 3! \text{ (arrange P, R, T)}
= 6 \times 4 \times 6 = \boxed{144}
$$

---

### ✅ **Final Answer**:

$$
\boxed{144}
$$

This is the total number of distinct seating arrangements of the 7 people such that no two of Pierre, Rosa, and Thomas sit adjacent to each other around the circular table.

Solution:
After Pierre sits, we can place Rosa either two seats from Pierre (that is, with one seat between them) or three seats from Pierre. We tackle these two cases separately:

Case 1: Rosa is two seats from Pierre. There are $2$ such seats. For either of these, there are then four empty seats in a row, and one empty seat between Rosa and Pierre. Thomas can sit in either of the middle two of the four empty seats in a row. So, there are $2\cdot 2 = 4$ ways to seat Rosa and Thomas in this case. There are then $4$ seats left, which the others can take in $4! = 24$ ways. So, there are $4\cdot 24 = 96$ seatings in this case.

Case 2: Rosa is three seats from Pierre (that is, there are $2$ seats between them). There are $2$ such seats. Thomas can't sit in either of the $2$ seats directly between them, but after Rosa sits, there are $3$ empty seats in a row still, and Thomas can only sit in the middle seat of these three. Once again, there are $4$ empty seats remaining, and the $4$ remaining people can sit in them in $4! = 24$ ways. So, we have $2\cdot 24 = 48$ seatings in this case.

Putting our two cases together gives a total of $96+48 = \boxed{144}$ seatings.