Explanation:
Okay, so I need to figure out how many ways 8 people can sit around a round table, but with the condition that Pierre, Rosa, and Thomas all want to sit together. And rotations are considered the same seating. Hmm, circular permutations can be a bit tricky because rotations are considered identical. Let me recall how circular permutations work.

For a circular table, the number of ways to arrange n people is (n-1)! instead of n! because rotations are considered the same. So, if there were no restrictions, the answer would be (8-1)! = 7! = 5040 ways. But here, there's a restriction: Pierre, Rosa, and Thomas must all sit together.

So, how do I handle that? I think when certain people need to be together, we can treat them as a single unit or "block." Then, we can arrange the blocks around the table and then consider the arrangements within the block. But since it's a circular table, I need to adjust for that.

Let me think. If Pierre, Rosa, and Thomas are sitting together, we can think of them as one combined entity. So instead of 8 individuals, we have this block plus the other 5 people. That makes 6 entities in total. But wait, in circular permutations, arranging n entities around a table is (n-1)! So, if we have 6 entities, the number of ways to arrange them would be (6-1)! = 5! = 120. But then, within the block of Pierre, Rosa, and Thomas, they can be arranged among themselves in different ways. How many ways can they be arranged? Since there are 3 people, it's 3! = 6.

Therefore, the total number of arrangements would be 5! * 3! = 120 * 6 = 720. But wait, is that all? Let me check if there's something I'm missing here.

Wait, hold on. When dealing with circular permutations and blocks, sometimes there's a different way to account for rotations. Let me verify.

Suppose we fix the position of the block to eliminate rotational symmetry. For example, in circular permutations, fixing one position is a common technique. If we fix the block of Pierre, Rosa, and Thomas in a specific position, then we can arrange the remaining people relative to this block. But since rotations are considered the same, fixing the block's position would account for that.

Alternatively, if we consider the block as a single entity, then arranging the 6 entities around the table would normally be (6-1)! = 5! ways. Then, multiplying by the internal arrangements of the block, which is 3!. So, 5! * 3! = 720.

But let me think again. Suppose we have n objects around a circular table. The formula is (n-1)! because rotations are considered the same. So, if we have 6 objects (the block and the other 5 people), then the number of arrangements is (6-1)! = 120. Then, multiply by the number of ways to arrange the people inside the block, which is 3! = 6. So, 120 * 6 = 720.

But wait, another way to think about it: if we first consider the total number of ways without considering rotations, then divide by the number of rotations. But maybe that complicates things. Let me try that approach.

If we consider the block as a single entity, then there are 6 entities. If we were to arrange them in a line, it would be 6! ways, and then multiply by 3! for the internal arrangements. But since it's a circular table, we need to adjust for rotations. In linear arrangements, the number is 6! * 3!, but for circular arrangements, it's (6-1)! * 3! = 5! * 3! = 720. So that seems consistent.

But let me check with a smaller number. Let's say there are 3 people, and they all want to sit together. How many ways? Well, if there are 3 people, and they all must sit together, then it's just 1 way (since it's a circular table). But according to the formula, n = 3, so treating them as a single block, we have 1 block, so (1-1)! * 3! = 0! * 6 = 1 * 6 = 6. Wait, that's not correct. Wait, if there are 3 people around a circular table, and they all must sit together, how many arrangements are there? Actually, since they are all sitting together, it's just 1 arrangement because rotations are considered the same. But according to the formula, it's (1-1)! * 3! = 1 * 6 = 6. Which is wrong. So, there's a problem here.

Wait, so maybe my approach is wrong? Let me think again.

If there are 3 people, and they all have to sit together, then since it's a circular table, there's only 1 way. But according to the formula, treating them as a single block, we have 1 block, so (1-1)! = 1, multiplied by 3! = 6. But that's not correct. So, clearly, my formula is wrong here.

What's the issue? Maybe when the number of blocks is 1, the formula (number of blocks -1)! * internal arrangements is not correct? Wait, but if there is only one block, then arranging it around the table... but there's only one block, so there's only 1 way to arrange it. But the internal arrangements would be 3! = 6. But since rotations are considered the same, but if there's only one block, rotating it doesn't change the arrangement. So, actually, the number of arrangements should be 3! / 3 = 2? Wait, no. Wait, if you have 3 people around a circular table, the number of distinct arrangements is (3-1)! = 2. But if they all have to sit together, which they are already doing, so the number of arrangements is 2. But according to my previous formula, it was 1 * 3! = 6. Which is wrong.

So, clearly, my approach is flawed. Where is the mistake?

Wait, maybe when we have a single block, the formula isn't (number of blocks -1)! * internal arrangements. Maybe I need to adjust for something else. Let me think again.

In the case of circular permutations with blocks, the standard approach is to consider the blocks as entities. For example, if you have k blocks, then the number of circular arrangements is (k-1)! multiplied by the product of the internal arrangements of each block.

But in the case where k=1, that would be (1-1)! * internal arrangements = 1 * 3! = 6. But in reality, when you have 3 people around a circular table, the number of distinct arrangements is 2, not 6. So, there's a contradiction here.

Wait, so why is that? Because when you have a single block, the formula (k-1)! * internal arrangements is not correct? Or is there a different way?

Wait, maybe the formula is different when the block is considered as a single entity. Let me think. If you have k entities, then the number of circular arrangements is (k-1)! So, if you have 1 entity, then (1-1)! = 1. But if that entity is a block of 3 people, then the number of arrangements is 1 * (number of internal arrangements). But in reality, the number of circular arrangements of 3 people is 2, which is (3-1)! = 2. So, if you have a single block of 3 people, then the number of circular arrangements is (number of internal arrangements) divided by something?

Wait, maybe the formula is that when you have blocks, you first fix the position of the block to account for circularity, then arrange the rest.

Alternatively, maybe the formula is (k-1)! * product of internal arrangements. But in the case of k=1, that would be 0! * 3! = 1 * 6 = 6, but the actual number is 2. So, that suggests that the formula is not correct.

Wait, so maybe the problem is that when you have a single block, you can't just use (k-1)! because the block itself is a group. Wait, but if you have a single block, then arranging it around the table is just 1 way, but the people inside the block can be arranged in 3! ways. However, in reality, the number of circular arrangements of 3 people is 2, which is (3-1)! = 2. So, if you have a block of 3 people, then the number of circular arrangements is 2, not 6. Therefore, there's a conflict here.

So, what is the correct way?

Perhaps the formula is that when you have blocks, you first consider the number of circular arrangements of the blocks, and then multiply by the internal arrangements. However, if the blocks are considered as entities, then the formula is (number of blocks -1)! * product of internal arrangements. But in the case of a single block, that would be (1-1)! * 3! = 1 * 6 = 6, but the actual number is 2. Therefore, there's a contradiction.

Wait, so maybe the formula is different? Let me check with another example. Suppose there are 2 people, and they must sit together. How many ways? Well, around a circular table, there's only 1 way. But according to the formula, if we have 1 block (the two people), then (1-1)! * 2! = 1 * 2 = 2. Which is wrong. So, again, the formula gives 2, but the correct answer is 1.

So, clearly, there's a problem here. Therefore, my initial approach is wrong.

So, why is this happening? Maybe because when you have a single block, the formula (k-1)! * internal arrangements is not correct?

Wait, perhaps the formula is (number of blocks)! * internal arrangements divided by the number of rotations? Wait, but how?

Alternatively, maybe when considering circular permutations with blocks, the formula is similar to linear permutations but adjusted for circularity.

In linear permutations, if we have k blocks, the number of arrangements is k! * product of internal arrangements. But for circular permutations, it's (k-1)! * product of internal arrangements. However, in the case where k=1, that would be 0! * internal arrangements = 1 * internal arrangements. But in reality, when you have a single block (which is the entire set of people), the number of circular arrangements is (n-1)! where n is the total number of people.

Wait, so if we have a single block of size n, then according to the formula, it would be (1-1)! * n! = 1 * n! = n! But the actual number of circular arrangements is (n-1)! So, there's a discrepancy here.

Therefore, the formula (k-1)! * product of internal arrangements is not correct when the blocks are not separate entities. Wait, but in the case where we have multiple blocks, maybe it's correct?

Wait, let me think again. Suppose we have two blocks. For example, two blocks of people. Let's say block A and block B. How many circular arrangements?

If we fix block A in a position to account for circularity, then block B can be arranged in 1 way (since it's the only other block), but the internal arrangements of block A and block B would be multiplied. So, total arrangements would be (number of internal arrangements of A) * (number of internal arrangements of B).

But if we use the formula (k-1)! * product of internal arrangements, where k=2, then (2-1)! * (internal A * internal B) = 1! * (internal A * internal B) = internal A * internal B. Which is the same as fixing one block and arranging the others.

But in the case of k=1, the formula gives 0! * internal arrangements = 1 * internal arrangements, but the actual number is (n-1)! where n is the total number of people in the block.

So, there's an inconsistency here. Therefore, maybe the formula is different when the block is the entire set?

Wait, perhaps the formula (k-1)! * product of internal arrangements is correct only when there are multiple blocks. But when there is only one block, we need to adjust it.

Alternatively, maybe the formula is correct, but in the case of a single block, the internal arrangements are not multiplied? Wait, no.

Wait, let me think of another way. Suppose we have n people, and we want to arrange them around a circular table. The number of ways is (n-1)!

If we consider them as a single block, then the number of ways is (1-1)! * n! = 1 * n! = n! which is different from (n-1)! So, clearly, there's a problem.

Therefore, the formula (k-1)! * product of internal arrangements is not correct.

So, what is the correct way?

Perhaps the correct approach is to first consider the total number of linear arrangements where the block is together, and then adjust for circularity.

For linear arrangements, if we have a block of 3 people, then we can treat them as a single entity. So, total entities: 6 (block + 5 others). The number of linear arrangements is 6! * 3! (since the block can be arranged internally in 3! ways).

But for circular arrangements, the number is different. In circular arrangements, the number of distinct arrangements is (number of linear arrangements) divided by the number of seats, which is n. Wait, is that correct?

Wait, in general, the number of circular arrangements is equal to the number of linear arrangements divided by n, because each circular arrangement corresponds to n linear arrangements (rotations).

So, if we have linear arrangements of the block and others, which is 6! * 3!, then the number of circular arrangements would be (6! * 3!) / 8.

Wait, but why 8? Because there are 8 seats.

Wait, let me check with the previous example. If we have 3 people, and we want to arrange them around a circular table. Using this method: treat them as a single block. Number of linear arrangements is 1! * 3! = 6. Then divide by 3 (number of people) to get circular arrangements: 6 / 3 = 2, which is correct.

Similarly, if we have 2 people, treating them as a single block: linear arrangements is 1! * 2! = 2. Divide by 2 (number of people) gives 1, which is correct.

So, in general, if we have a block of size m, and the rest of the people, then the number of circular arrangements would be (number of linear arrangements) / total number of people.

So, in our original problem, we have 8 people, with a block of 3. So, treating the block as a single entity, we have 6 entities. The number of linear arrangements is 6! * 3! (arrangements of the entities times internal arrangements of the block). Then, to convert to circular arrangements, divide by 8 (the total number of people).

Therefore, total number of circular arrangements is (6! * 3!) / 8.

Let me compute that. 6! is 720, 3! is 6, so 720 * 6 = 4320. Then divide by 8: 4320 / 8 = 540.

But earlier, my initial approach gave 5! * 3! = 120 * 6 = 720. Which is different.

So, which one is correct?

Let me check with the previous example. If we have 3 people, and we use this formula: (number of linear arrangements) / n.

Number of linear arrangements: treating them as a single block, which is 1! * 3! = 6. Divide by 3 (n=3) gives 2, which is correct.

Alternatively, if I use the formula (k-1)! * internal arrangements, where k is the number of blocks. For k=1, (1-1)! * 3! = 1 * 6 = 6, which is wrong. But with the other approach, it's 6 / 3 = 2.

So, the correct formula seems to be (number of linear arrangements) / n.

But why is that?

Because in circular arrangements, each unique arrangement is counted n times in the linear arrangements (once for each rotation). Therefore, to get the number of circular arrangements, we divide by n.

Therefore, in our original problem, the number of circular arrangements where the 3 people are together is (6! * 3!) / 8.

But let me check with another example. Suppose we have 4 people, and 2 of them must sit together. How many circular arrangements?

Using the formula: treat the 2 people as a block. So, number of entities: 3 (block + 2 others). Number of linear arrangements: 3! * 2! = 12. Divide by 4 (total people) gives 12 / 4 = 3.

Alternatively, let's compute manually.

Total number of circular arrangements where two specific people are together.

First, fix the position of the block to eliminate rotations. Let's fix the block of two people. Then, the remaining two people can be arranged in 2! ways. But the block itself can be arranged in 2! ways. However, since we fixed the block's position, we don't need to consider rotations. Wait, but fixing the block's position would mean that the number of arrangements is 2! * 2! = 4? But according to the formula, it's 3.

Wait, there's a contradiction here.

Wait, let me think again. If we have 4 people: A, B, C, D. Suppose A and B must sit together. How many circular arrangements?

First, fix A's position to eliminate rotational symmetry. Since it's a circular table, we can fix A at a position. Then, B can be either to the left or right of A. So, 2 choices. Then, the remaining two people, C and D, can be arranged in the remaining two seats in 2! = 2 ways. So total arrangements: 2 * 2 = 4.

But according to the formula (number of linear arrangements)/n: treating A and B as a block, number of entities is 3 (block, C, D). Number of linear arrangements: 3! * 2! = 12. Divide by 4 gives 3. But manual count gives 4.

So, discrepancy again.

Hmm, so which is correct?

Let me list all possible arrangements.

Fix A at a position. Then, B can be to the left or right.

Case 1: B is to the right of A. Then, the seats are A, B, ?, ?. The remaining seats are clockwise from B: seat 3 and seat 4. These can be filled by C and D in 2 ways: C then D, or D then C.

Case 2: B is to the left of A. Then, the seats are A, ?, ?, B. The remaining seats are seat 2 and seat 3. These can be filled by C and D in 2 ways.

So total of 2 * 2 = 4 arrangements.

But according to the formula (number of linear arrangements)/n: 3! * 2! / 4 = 12 / 4 = 3.

So, discrepancy. Therefore, the formula is not correct?

Alternatively, maybe my manual count is wrong?

Wait, let me think again. If we fix A's position, then the number of arrangements is 2 * 2 = 4. But if we don't fix positions, how many are there?

In circular arrangements, fixing A's position is a way to count, but if we don't fix positions, the total number is (number of circular arrangements).

But according to the formula, it's 3. But manual count gives 4.

Wait, maybe the formula is wrong.

Alternatively, maybe the formula should be (number of linear arrangements) / n, but in this case, the number of linear arrangements where A and B are together is 2 * 3! = 12? Wait, no.

Wait, if we consider linear arrangements where A and B are together, we can treat them as a block. So, number of linear arrangements is 3! * 2! = 12. But in circular arrangements, how does that translate?

But in circular arrangements, the number is different.

Alternatively, maybe the formula is (number of linear arrangements) / n, but in this case, the number of circular arrangements is 12 / 4 = 3. But manual count says 4.

So, which is correct?

Wait, let me think of all possible circular arrangements where A and B are together.

In circular table with 4 people, the number of distinct arrangements where A and B are adjacent.

Each such arrangement can be represented as A-B-C-D, A-B-D-C, A-C-B-D, A-D-B-C, etc. But considering rotations and reflections? Wait, no, the problem says two seatings are the same if one is a rotation of the other. Reflections are considered different.

So, let's fix A's position. Then, B can be to the left or right.

If A is fixed, then B can be in two positions. Then, the rest can be arranged.

But if we don't fix positions, how many are there?

Each unique circular arrangement where A and B are adjacent can be rotated so that A is in a specific position. So, fixing A's position, we have 2 possibilities for B, and then 2! for the rest. So total 4.

But according to the formula (number of linear arrangements)/n, which was 12 / 4 = 3.

So, there's a contradiction. Therefore, my formula is wrong.

Alternatively, maybe the formula is (number of linear arrangements) / n, but in this case, the number of linear arrangements where A and B are together is not 3! * 2! = 12.

Wait, let me compute the number of linear arrangements where A and B are together.

In linear arrangements, the number is 2 * 3! = 12? Wait, no.

Wait, if we have 4 people, and A and B must be together. Treat them as a block. Then, the number of linear arrangements is 3! * 2! = 12.

But in circular arrangements, how does that translate?

But in circular arrangements, the number is different.

Alternatively, maybe the formula is (number of linear arrangements) / n, but in this case, the number of circular arrangements is 12 / 4 = 3. But manual count says 4.

So, which is correct?

Wait, let me think of all possible circular arrangements where A and B are adjacent.

Let me list them:

1. A-B-C-D

2. A-B-D-C

3. A-C-B-D

4. A-D-B-C

But since it's a circular table, some of these might be rotations of each other.

Wait, if we fix A's position, then these are distinct. But if we don't fix positions, then some might be rotations.

Wait, for example, arrangement 1: A-B-C-D. If we rotate it so that B is first, we get B-C-D-A. Is that considered the same? No, because the problem says two seatings are the same if one is a rotation of the other. So, if we consider rotations as the same, then A-B-C-D is the same as B-C-D-A, etc.

Therefore, if we fix A's position, we are effectively counting each unique arrangement once.

But if we don't fix positions, how many unique arrangements are there?

Each unique arrangement can be rotated so that A is in a specific position. Therefore, the number of unique arrangements is equal to the number of arrangements with A fixed.

Therefore, if fixing A's position gives 4 arrangements, then the total number of unique circular arrangements is 4.

But according to the formula (number of linear arrangements)/n, which was 12 / 4 = 3.

So, there's a contradiction. Therefore, my formula is wrong.

So, what is the correct formula?

Perhaps the formula is different. Maybe when considering circular arrangements with blocks, we need to use (k-1)! * internal arrangements, but in the case where the block is part of the total.

Wait, in the case of 4 people with A and B together, treating them as a block, we have 3 entities: [AB], C, D.

Number of circular arrangements of these 3 entities is (3-1)! = 2. Then, multiply by internal arrangements of the block: 2! = 2. So total is 2 * 2 = 4, which matches the manual count.

Ah! So, in this case, the formula (k-1)! * product of internal arrangements works.

Earlier, when I tried with k=1, it didn't work, but maybe that's a special case?

Wait, let's check with k=1. If we have 3 people as a single block, then k=1. So, (1-1)! * internal arrangements = 1 * 3! = 6. But the actual number of circular arrangements is 2.

But in the case of k=3, with 3 entities, it works.

Wait, so why is there a discrepancy?

Wait, maybe when the block is the entire set, the formula is different?

Wait, if we have 3 people as a single block, then the number of circular arrangements is (number of circular arrangements of the block). But the block is 3 people, so the number of circular arrangements is (3-1)! = 2. But according to the formula (k-1)! * internal arrangements, where k=1, it's 0! * 3! = 6.

So, clearly, the formula is not applicable when the block is the entire set.

But in the case of 4 people with a block of 2, the formula works.

So, what's the difference?

Ah! Maybe when the block is not the entire set, the formula (k-1)! * product of internal arrangements is correct. But when the block is the entire set, it's different.

But why?

Wait, in the case of the block being the entire set, the number of circular arrangements is (n-1)! where n is the total number of people. But according to the formula (k-1)! * product of internal arrangements, where k=1, it's 0! * n! = n! which is different.

Therefore, the formula works when there are multiple blocks, but not when there is only one block.

But in the case of 4 people with a block of 2, we have k=3 entities (block, C, D). So, k=3, and the formula (k-1)! * internal arrangements = 2! * 2! = 4, which matches the manual count.

So, maybe the formula is correct when there are multiple blocks.

Therefore, going back to the original problem.

Original problem: 8 people, with 3 people (Pierre, Rosa, Thomas) sitting together.

So, we have 1 block (P-R-T) and 5 other people. So, total of 6 entities.

Therefore, number of circular arrangements is (6-1)! * internal arrangements of the block = 5! * 3! = 120 * 6 = 720.

But earlier, when I tried with the formula (number of linear arrangements)/n, I got 540.

But in the case of 4 people with a block of 2, the formula (k-1)! * internal arrangements worked.

So, why the discrepancy?

Wait, in the case of 4 people with a block of 2, the number of circular arrangements is (k-1)! * internal arrangements = 2! * 2! = 4, which matches the manual count.

But if I use the formula (number of linear arrangements)/n, where number of linear arrangements is 3! * 2! = 12, divided by n=4, gives 3, which is wrong.

Therefore, the formula (k-1)! * internal arrangements is correct, and the other formula is wrong.

So, why did the formula (number of linear arrangements)/n give a wrong answer?

Because when we treat the block as an entity, the number of linear arrangements is k! * product of internal arrangements, where k is the number of entities. But for circular arrangements, it's (k-1)! * product of internal arrangements.

Therefore, the correct formula is (k-1)! * product of internal arrangements.

But in the case of k=1, this formula gives (1-1)! * internal arrangements = 1 * n! which is wrong. But in reality, when k=1, the number of circular arrangements is (n-1)!

So, there's an inconsistency here.

But in the case where we have multiple blocks, the formula works.

Therefore, maybe the formula is correct when there are multiple blocks, and when there is only one block, we need to adjust.

But in our original problem, we have multiple blocks (the block of 3 and 5 other individuals). Therefore, the formula (k-1)! * product of internal arrangements is correct.

Therefore, going back to the original problem:

Number of circular arrangements = (6-1)! * 3! = 5! * 6 = 120 * 6 = 720.

But earlier, when I thought of fixing the block, I thought of fixing the block's position to eliminate rotations.

Alternatively, another way to think about it:

In circular permutations, fixing one position is a common technique. So, if we fix the block of Pierre, Rosa, and Thomas in a specific position, then we can arrange the remaining 5 people in the remaining seats.

But since the block is fixed, we don't have to worry about rotations.

However, within the block, Pierre, Rosa, and Thomas can be arranged in 3! ways.

The remaining 5 people can be arranged in 5! ways.

Therefore, total number of arrangements is 3! * 5! = 6 * 120 = 720.

Which matches the previous result.

But why did the other approach give a different answer?

Because when I considered the formula (number of linear arrangements)/n, I think I was making a mistake in how I calculated the linear arrangements.

Wait, if we fix the block's position, then the number of linear arrangements is 5! * 3! = 720. But since we fixed the block's position, we don't need to divide by anything.

But in circular arrangements, fixing one position is equivalent to considering the number of circular arrangements as (number of linear arrangements with one position fixed).

Therefore, the answer is 5! * 3! = 720.

Therefore, the correct answer is 720.

But then why did the previous example with 4 people and a block of 2 give a different result when I tried the formula (number of linear arrangements)/n?

Because in that case, if I fix the block's position, then the number of arrangements is 2! * 2! = 4, which matches the manual count.

But if I use the formula (number of linear arrangements)/n, where number of linear arrangements is 3! * 2! = 12, divided by n=4, gives 3, which is wrong.

Therefore, the formula (number of linear arrangements)/n is not correct.

Therefore, the correct approach is to treat the block as an entity, calculate (k-1)! * product of internal arrangements, where k is the number of entities.

Therefore, in the original problem, the answer is 5! * 3! = 720.

But then why did the example with 3 people give a discrepancy?

Because when k=1, the formula gives (1-1)! * internal arrangements = 1 * 3! = 6, but the actual number of circular arrangements is 2.

But in that case, the block is the entire set, so maybe the formula is not applicable?

But in our original problem, the block is not the entire set.

Therefore, the formula works when there are multiple blocks.

Therefore, the answer is 720.

But let me check with another example.

Suppose we have 5 people, and 2 of them must sit together.

Number of circular arrangements:

Treat the 2 as a block. So, k=4 entities (block + 3 others).

Number of circular arrangements: (4-1)! * 2! = 6 * 2 = 12.

Alternatively, fix the block's position. Then, the remaining 3 people can be arranged in 3! ways, and the block can be arranged in 2! ways. Total: 3! * 2! = 12.

Which is correct.

Alternatively, if I use the formula (number of linear arrangements)/n:

Number of linear arrangements is 4! * 2! = 24 * 2 = 48. Divide by 5 gives 9.6, which is not an integer. So, clearly wrong.

Therefore, the formula (number of linear arrangements)/n is not correct.

Therefore, the correct formula is (k-1)! * product of internal arrangements.

Therefore, returning to the original problem, the answer is 5! * 3! = 720.

But let me check with another approach.

Imagine that we have 8 seats around a table. We want Pierre, Rosa, and Thomas to sit together.

First, consider the number of ways to seat them together.

First, we can think of the number of ways to choose a block of 3 consecutive seats. In a circular table, the number of ways to choose a block of 3 consecutive seats is 8 (since there are 8 seats, each seat can be the start of a block of 3).

But since rotations are considered the same, we need to be careful.

Wait, but if we fix the block's position, then we can avoid overcounting.

Alternatively, if we don't fix positions, the number of ways to arrange the block is 8 (positions for the block) * 3! (arrangements within the block) * 5! (arrangements of the others). But since rotations are considered the same, we have overcounted by a factor of 8.

Therefore, total number of arrangements is (8 * 3! * 5!) / 8 = 3! * 5! = 720.

Ah! So this approach gives the same result as before.

Therefore, this confirms that the answer is 720.

Therefore, the initial formula (k-1)! * product of internal arrangements is correct, and the confusion with the single block case is because when the block is the entire set, the formula doesn't apply, but in our problem, we have multiple blocks.

Therefore, the answer is 5! * 3! = 720.

But let me check with the example of 3 people. If we have 3 people, and we want them to sit together.

Number of circular arrangements:

Using this approach: number of ways to choose a block of 3 seats: 3 (since circular table, number of blocks of 3 consecutive seats is 3). Then, arrangements: 3 * 3! / 3 = 3! = 6. But the actual number of circular arrangements is 2.

Wait, discrepancy again.

But why?

Because when we have the entire set as a block, the number of ways to choose the block is equal to the number of seats, but since rotations are considered the same, we have overcounted.

Wait, in this case, if we have 3 people, and we consider the number of ways to seat them together:

Number of ways to choose the block: 3 (positions). Then, arrangements within the block: 3!. Then, divide by 3 (number of rotations) to get 3! / 3 = 2, which is correct.

Ah! So, in this case, the formula is (number of ways to choose the block * internal arrangements) / n.

So, in general, the formula is:

Number of circular arrangements = (number of ways to choose the block * internal arrangements of the block * arrangements of the rest) / n.

But in our original problem, the number of ways to choose the block is 8 (since there are 8 seats, and each block of 3 consecutive seats can be placed in 8 different positions).

But since rotations are considered the same, we need to divide by n=8.

Therefore, total number of arrangements is (8 * 3! * 5!) / 8 = 3! * 5! = 720.

Which matches the previous result.

Therefore, this formula works.

So, in the case of 3 people:

Number of ways to choose the block: 3.

Internal arrangements: 3!.

Arrangements of the rest: since there are no others, it's 1.

Total: (3 * 3! * 1) / 3 = 3! = 6. But the actual number of circular arrangements is 2.

Wait, discrepancy again.

Wait, what's wrong here?

Ah! Because when there are no other people, the arrangements of the rest is 1, but we also need to consider that the block is the entire set.

Wait, but if we have 3 people, and we want to arrange them around a table with the block of 3 people, then the number of circular arrangements is (number of ways to choose the block * internal arrangements) / n.

But number of ways to choose the block is 3, internal arrangements is 3!, divided by n=3 gives 3! = 6. But the actual number is 2.

So, why is this?

Because when we choose the block, we are considering different positions, but in circular arrangements, those are considered the same.

Wait, but if we have 3 people, and we fix the block's position, then the number of arrangements is 3! / 3 = 2.

But according to the formula (number of ways to choose the block * internal arrangements) / n, it's (3 * 3!) / 3 = 6.

So, why is this different?

Because when we choose the block, we are considering different positions, but in reality, those positions are rotations of each other.

Therefore, the formula (number of ways to choose the block * internal arrangements) / n is not correct.

But in the case of the original problem, where there are other people, the formula works.

Wait, in the original problem, when we have other people, the arrangements of the rest are different for each block position.

Wait, no. If we have other people, then choosing different positions for the block would lead to different arrangements.

But in the case of 3 people, there are no other people, so choosing different positions for the block doesn't change the arrangement.

Therefore, the formula (number of ways to choose the block * internal arrangements) / n is not correct in general.

But in the case where there are other people, it works?

Wait, in the case of 4 people with a block of 2, using this formula:

Number of ways to choose the block: 4 (since there are 4 seats, each block of 2 consecutive seats can be placed in 4 positions).

Internal arrangements: 2!.

Arrangements of the rest: 2! (since there are 2 other people).

Total: (4 * 2! * 2!) / 4 = 2! * 2! = 4, which matches the manual count.

So, in this case, it works.

But in the case of 3 people, it doesn't.

So, why?

Because when there are other people, the arrangements of the rest are different for each block position, so dividing by n accounts for the overcounting.

But when there are no other people, the arrangements of the rest are 1, and dividing by n gives an incorrect result.

Therefore, the formula works when there are other people, but not when the block is the entire set.

Therefore, in our original problem, since there are other people, the formula works.

Therefore, the answer is 720.

Therefore, after all this thinking, I believe the correct answer is 720.

**Final Answer**
The number of ways is \boxed{720}.

To determine the number of ways 8 people can sit around a round table such that **Pierre, Rosa, and Thomas all sit together**, we need to carefully account for both the **circular nature of the table** and the **restriction that three specific people must be adjacent**.

---

### Step 1: Treat the Group as a Single Block

Since **Pierre, Rosa, and Thomas** must sit together, we can treat them as a **single block** or unit. This reduces the problem from arranging 8 individuals to arranging **6 entities**: the block of 3 people and the other 5 individuals.

---

### Step 2: Circular Permutations of the 6 Entities

In a **circular arrangement**, the number of distinct ways to arrange $ n $ entities is:

$$
(n - 1)!
$$

So, for 6 entities (the block and the 5 others), the number of circular arrangements is:

$$
(6 - 1)! = 5! = 120
$$

---

### Step 3: Internal Arrangements Within the Block

Within the block of 3 people (Pierre, Rosa, and Thomas), they can be arranged among themselves in:

$$
3! = 6
$$

ways.

---

### Step 4: Total Number of Arrangements

To find the total number of valid arrangements, we multiply the number of circular arrangements of the 6 entities by the number of internal arrangements within the block:

$$
5! \times 3! = 120 \times 6 = 720
$$

---

### Final Answer

$$
\boxed{720}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways 8 people can sit around a round table, but with the condition that Pierre, Rosa, and Thomas all want to sit together. And rotations are considered the same seating. Hmm, circular permutations can be a bit tricky because rotations are considered identical. Let me recall how circular permutations work.

For a circular table, the number of ways to arrange n people is (n-1)! instead of n! because rotations are considered the same. So, if there were no restrictions, the answer would be (8-1)! = 7! = 5040 ways. But here, there's a restriction: Pierre, Rosa, and Thomas must all sit together.

So, how do I handle that? I think when certain people need to be together, we can treat them as a single unit or "block." Then, we can arrange the blocks around the table and then consider the arrangements within the block. But since it's a circular table, I need to adjust for that.

Let me think. If Pierre, Rosa, and Thomas are sitting together, we can think of them as one combined entity. So instead of 8 individuals, we have this block plus the other 5 people. That makes 6 entities in total. But wait, in circular permutations, arranging n entities around a table is (n-1)! So, if we have 6 entities, the number of ways to arrange them would be (6-1)! = 5! = 120. But then, within the block of Pierre, Rosa, and Thomas, they can be arranged among themselves in different ways. How many ways can they be arranged? Since there are 3 people, it's 3! = 6.

Therefore, the total number of arrangements would be 5! * 3! = 120 * 6 = 720. But wait, is that all? Let me check if there's something I'm missing here.

Wait, hold on. When dealing with circular permutations and blocks, sometimes there's a different way to account for rotations. Let me verify.

Suppose we fix the position of the block to eliminate rotational symmetry. For example, in circular permutations, fixing one position is a common technique. If we fix the block of Pierre, Rosa, and Thomas in a specific position, then we can arrange the remaining people relative to this block. But since rotations are considered the same, fixing the block's position would account for that.

Alternatively, if we consider the block as a single entity, then arranging the 6 entities around the table would normally be (6-1)! = 5! ways. Then, multiplying by the internal arrangements of the block, which is 3!. So, 5! * 3! = 720.

But let me think again. Suppose we have n objects around a circular table. The formula is (n-1)! because rotations are considered the same. So, if we have 6 objects (the block and the other 5 people), then the number of arrangements is (6-1)! = 120. Then, multiply by the number of ways to arrange the people inside the block, which is 3! = 6. So, 120 * 6 = 720.

But wait, another way to think about it: if we first consider the total number of ways without considering rotations, then divide by the number of rotations. But maybe that complicates things. Let me try that approach.

If we consider the block as a single entity, then there are 6 entities. If we were to arrange them in a line, it would be 6! ways, and then multiply by 3! for the internal arrangements. But since it's a circular table, we need to adjust for rotations. In linear arrangements, the number is 6! * 3!, but for circular arrangements, it's (6-1)! * 3! = 5! * 3! = 720. So that seems consistent.

But let me check with a smaller number. Let's say there are 3 people, and they all want to sit together. How many ways? Well, if there are 3 people, and they all must sit together, then it's just 1 way (since it's a circular table). But according to the formula, n = 3, so treating them as a single block, we have 1 block, so (1-1)! * 3! = 0! * 6 = 1 * 6 = 6. Wait, that's not correct. Wait, if there are 3 people around a circular table, and they all must sit together, how many arrangements are there? Actually, since they are all sitting together, it's just 1 arrangement because rotations are considered the same. But according to the formula, it's (1-1)! * 3! = 1 * 6 = 6. Which is wrong. So, there's a problem here.

Wait, so maybe my approach is wrong? Let me think again.

If there are 3 people, and they all have to sit together, then since it's a circular table, there's only 1 way. But according to the formula, treating them as a single block, we have 1 block, so (1-1)! = 1, multiplied by 3! = 6. But that's not correct. So, clearly, my formula is wrong here.

What's the issue? Maybe when the number of blocks is 1, the formula (number of blocks -1)! * internal arrangements is not correct? Wait, but if there is only one block, then arranging it around the table... but there's only one block, so there's only 1 way to arrange it. But the internal arrangements would be 3! = 6. But since rotations are considered the same, but if there's only one block, rotating it doesn't change the arrangement. So, actually, the number of arrangements should be 3! / 3 = 2? Wait, no. Wait, if you have 3 people around a circular table, the number of distinct arrangements is (3-1)! = 2. But if they all have to sit together, which they are already doing, so the number of arrangements is 2. But according to my previous formula, it was 1 * 3! = 6. Which is wrong.

So, clearly, my approach is flawed. Where is the mistake?

Wait, maybe when we have a single block, the formula isn't (number of blocks -1)! * internal arrangements. Maybe I need to adjust for something else. Let me think again.

In the case of circular permutations with blocks, the standard approach is to consider the blocks as entities. For example, if you have k blocks, then the number of circular arrangements is (k-1)! multiplied by the product of the internal arrangements of each block.

But in the case where k=1, that would be (1-1)! * internal arrangements = 1 * 3! = 6. But in reality, when you have 3 people around a circular table, the number of distinct arrangements is 2, not 6. So, there's a contradiction here.

Wait, so why is that? Because when you have a single block, the formula (k-1)! * internal arrangements is not correct? Or is there a different way?

Wait, maybe the formula is different when the block is considered as a single entity. Let me think. If you have k entities, then the number of circular arrangements is (k-1)! So, if you have 1 entity, then (1-1)! = 1. But if that entity is a block of 3 people, then the number of arrangements is 1 * (number of internal arrangements). But in reality, the number of circular arrangements of 3 people is 2, which is (3-1)! = 2. So, if you have a single block of 3 people, then the number of circular arrangements is (number of internal arrangements) divided by something?

Wait, maybe the formula is that when you have blocks, you first fix the position of the block to account for circularity, then arrange the rest.

Alternatively, maybe the formula is (k-1)! * product of internal arrangements. But in the case of k=1, that would be 0! * 3! = 1 * 6 = 6, but the actual number is 2. So, that suggests that the formula is not correct.

Wait, so maybe the problem is that when you have a single block, you can't just use (k-1)! because the block itself is a group. Wait, but if you have a single block, then arranging it around the table is just 1 way, but the people inside the block can be arranged in 3! ways. However, in reality, the number of circular arrangements of 3 people is 2, which is (3-1)! = 2. So, if you have a block of 3 people, then the number of circular arrangements is 2, not 6. Therefore, there's a conflict here.

So, what is the correct way?

Perhaps the formula is that when you have blocks, you first consider the number of circular arrangements of the blocks, and then multiply by the internal arrangements. However, if the blocks are considered as entities, then the formula is (number of blocks -1)! * product of internal arrangements. But in the case of a single block, that would be (1-1)! * 3! = 1 * 6 = 6, but the actual number is 2. Therefore, there's a contradiction.

Wait, so maybe the formula is different? Let me check with another example. Suppose there are 2 people, and they must sit together. How many ways? Well, around a circular table, there's only 1 way. But according to the formula, if we have 1 block (the two people), then (1-1)! * 2! = 1 * 2 = 2. Which is wrong. So, again, the formula gives 2, but the correct answer is 1.

So, clearly, there's a problem here. Therefore, my initial approach is wrong.

So, why is this happening? Maybe because when you have a single block, the formula (k-1)! * internal arrangements is not correct?

Wait, perhaps the formula is (number of blocks)! * internal arrangements divided by the number of rotations? Wait, but how?

Alternatively, maybe when considering circular permutations with blocks, the formula is similar to linear permutations but adjusted for circularity.

In linear permutations, if we have k blocks, the number of arrangements is k! * product of internal arrangements. But for circular permutations, it's (k-1)! * product of internal arrangements. However, in the case where k=1, that would be 0! * internal arrangements = 1 * internal arrangements. But in reality, when you have a single block (which is the entire set of people), the number of circular arrangements is (n-1)! where n is the total number of people.

Wait, so if we have a single block of size n, then according to the formula, it would be (1-1)! * n! = 1 * n! = n! But the actual number of circular arrangements is (n-1)! So, there's a discrepancy here.

Therefore, the formula (k-1)! * product of internal arrangements is not correct when the blocks are not separate entities. Wait, but in the case where we have multiple blocks, maybe it's correct?

Wait, let me think again. Suppose we have two blocks. For example, two blocks of people. Let's say block A and block B. How many circular arrangements?

If we fix block A in a position to account for circularity, then block B can be arranged in 1 way (since it's the only other block), but the internal arrangements of block A and block B would be multiplied. So, total arrangements would be (number of internal arrangements of A) * (number of internal arrangements of B).

But if we use the formula (k-1)! * product of internal arrangements, where k=2, then (2-1)! * (internal A * internal B) = 1! * (internal A * internal B) = internal A * internal B. Which is the same as fixing one block and arranging the others.

But in the case of k=1, the formula gives 0! * internal arrangements = 1 * internal arrangements, but the actual number is (n-1)! where n is the total number of people in the block.

So, there's an inconsistency here. Therefore, maybe the formula is different when the block is the entire set?

Wait, perhaps the formula (k-1)! * product of internal arrangements is correct only when there are multiple blocks. But when there is only one block, we need to adjust it.

Alternatively, maybe the formula is correct, but in the case of a single block, the internal arrangements are not multiplied? Wait, no.

Wait, let me think of another way. Suppose we have n people, and we want to arrange them around a circular table. The number of ways is (n-1)!

If we consider them as a single block, then the number of ways is (1-1)! * n! = 1 * n! = n! which is different from (n-1)! So, clearly, there's a problem.

Therefore, the formula (k-1)! * product of internal arrangements is not correct.

So, what is the correct way?

Perhaps the correct approach is to first consider the total number of linear arrangements where the block is together, and then adjust for circularity.

For linear arrangements, if we have a block of 3 people, then we can treat them as a single entity. So, total entities: 6 (block + 5 others). The number of linear arrangements is 6! * 3! (since the block can be arranged internally in 3! ways).

But for circular arrangements, the number is different. In circular arrangements, the number of distinct arrangements is (number of linear arrangements) divided by the number of seats, which is n. Wait, is that correct?

Wait, in general, the number of circular arrangements is equal to the number of linear arrangements divided by n, because each circular arrangement corresponds to n linear arrangements (rotations).

So, if we have linear arrangements of the block and others, which is 6! * 3!, then the number of circular arrangements would be (6! * 3!) / 8.

Wait, but why 8? Because there are 8 seats.

Wait, let me check with the previous example. If we have 3 people, and we want to arrange them around a circular table. Using this method: treat them as a single block. Number of linear arrangements is 1! * 3! = 6. Then divide by 3 (number of people) to get circular arrangements: 6 / 3 = 2, which is correct.

Similarly, if we have 2 people, treating them as a single block: linear arrangements is 1! * 2! = 2. Divide by 2 (number of people) gives 1, which is correct.

So, in general, if we have a block of size m, and the rest of the people, then the number of circular arrangements would be (number of linear arrangements) / total number of people.

So, in our original problem, we have 8 people, with a block of 3. So, treating the block as a single entity, we have 6 entities. The number of linear arrangements is 6! * 3! (arrangements of the entities times internal arrangements of the block). Then, to convert to circular arrangements, divide by 8 (the total number of people).

Therefore, total number of circular arrangements is (6! * 3!) / 8.

Let me compute that. 6! is 720, 3! is 6, so 720 * 6 = 4320. Then divide by 8: 4320 / 8 = 540.

But earlier, my initial approach gave 5! * 3! = 120 * 6 = 720. Which is different.

So, which one is correct?

Let me check with the previous example. If we have 3 people, and we use this formula: (number of linear arrangements) / n.

Number of linear arrangements: treating them as a single block, which is 1! * 3! = 6. Divide by 3 (n=3) gives 2, which is correct.

Alternatively, if I use the formula (k-1)! * internal arrangements, where k is the number of blocks. For k=1, (1-1)! * 3! = 1 * 6 = 6, which is wrong. But with the other approach, it's 6 / 3 = 2.

So, the correct formula seems to be (number of linear arrangements) / n.

But why is that?

Because in circular arrangements, each unique arrangement is counted n times in the linear arrangements (once for each rotation). Therefore, to get the number of circular arrangements, we divide by n.

Therefore, in our original problem, the number of circular arrangements where the 3 people are together is (6! * 3!) / 8.

But let me check with another example. Suppose we have 4 people, and 2 of them must sit together. How many circular arrangements?

Using the formula: treat the 2 people as a block. So, number of entities: 3 (block + 2 others). Number of linear arrangements: 3! * 2! = 12. Divide by 4 (total people) gives 12 / 4 = 3.

Alternatively, let's compute manually.

Total number of circular arrangements where two specific people are together.

First, fix the position of the block to eliminate rotations. Let's fix the block of two people. Then, the remaining two people can be arranged in 2! ways. But the block itself can be arranged in 2! ways. However, since we fixed the block's position, we don't need to consider rotations. Wait, but fixing the block's position would mean that the number of arrangements is 2! * 2! = 4? But according to the formula, it's 3.

Wait, there's a contradiction here.

Wait, let me think again. If we have 4 people: A, B, C, D. Suppose A and B must sit together. How many circular arrangements?

First, fix A's position to eliminate rotational symmetry. Since it's a circular table, we can fix A at a position. Then, B can be either to the left or right of A. So, 2 choices. Then, the remaining two people, C and D, can be arranged in the remaining two seats in 2! = 2 ways. So total arrangements: 2 * 2 = 4.

But according to the formula (number of linear arrangements)/n: treating A and B as a block, number of entities is 3 (block, C, D). Number of linear arrangements: 3! * 2! = 12. Divide by 4 gives 3. But manual count gives 4.

So, discrepancy again.

Hmm, so which is correct?

Let me list all possible arrangements.

Fix A at a position. Then, B can be to the left or right.

Case 1: B is to the right of A. Then, the seats are A, B, ?, ?. The remaining seats are clockwise from B: seat 3 and seat 4. These can be filled by C and D in 2 ways: C then D, or D then C.

Case 2: B is to the left of A. Then, the seats are A, ?, ?, B. The remaining seats are seat 2 and seat 3. These can be filled by C and D in 2 ways.

So total of 2 * 2 = 4 arrangements.

But according to the formula (number of linear arrangements)/n: 3! * 2! / 4 = 12 / 4 = 3.

So, discrepancy. Therefore, the formula is not correct?

Alternatively, maybe my manual count is wrong?

Wait, let me think again. If we fix A's position, then the number of arrangements is 2 * 2 = 4. But if we don't fix positions, how many are there?

In circular arrangements, fixing A's position is a way to count, but if we don't fix positions, the total number is (number of circular arrangements).

But according to the formula, it's 3. But manual count gives 4.

Wait, maybe the formula is wrong.

Alternatively, maybe the formula should be (number of linear arrangements) / n, but in this case, the number of linear arrangements where A and B are together is 2 * 3! = 12? Wait, no.

Wait, if we consider linear arrangements where A and B are together, we can treat them as a block. So, number of linear arrangements is 3! * 2! = 12. But in circular arrangements, how does that translate?

But in circular arrangements, the number is different.

Alternatively, maybe the formula is (number of linear arrangements) / n, but in this case, the number of circular arrangements is 12 / 4 = 3. But manual count says 4.

So, which is correct?

Wait, let me think of all possible circular arrangements where A and B are together.

In circular table with 4 people, the number of distinct arrangements where A and B are adjacent.

Each such arrangement can be represented as A-B-C-D, A-B-D-C, A-C-B-D, A-D-B-C, etc. But considering rotations and reflections? Wait, no, the problem says two seatings are the same if one is a rotation of the other. Reflections are considered different.

So, let's fix A's position. Then, B can be to the left or right.

If A is fixed, then B can be in two positions. Then, the rest can be arranged.

But if we don't fix positions, how many are there?

Each unique circular arrangement where A and B are adjacent can be rotated so that A is in a specific position. So, fixing A's position, we have 2 possibilities for B, and then 2! for the rest. So total 4.

But according to the formula (number of linear arrangements)/n, which was 12 / 4 = 3.

So, there's a contradiction. Therefore, my formula is wrong.

Alternatively, maybe the formula is (number of linear arrangements) / n, but in this case, the number of linear arrangements where A and B are together is not 3! * 2! = 12.

Wait, let me compute the number of linear arrangements where A and B are together.

In linear arrangements, the number is 2 * 3! = 12? Wait, no.

Wait, if we have 4 people, and A and B must be together. Treat them as a block. Then, the number of linear arrangements is 3! * 2! = 12.

But in circular arrangements, how does that translate?

But in circular arrangements, the number is different.

Alternatively, maybe the formula is (number of linear arrangements) / n, but in this case, the number of circular arrangements is 12 / 4 = 3. But manual count says 4.

So, which is correct?

Wait, let me think of all possible circular arrangements where A and B are adjacent.

Let me list them:

1. A-B-C-D

2. A-B-D-C

3. A-C-B-D

4. A-D-B-C

But since it's a circular table, some of these might be rotations of each other.

Wait, if we fix A's position, then these are distinct. But if we don't fix positions, then some might be rotations.

Wait, for example, arrangement 1: A-B-C-D. If we rotate it so that B is first, we get B-C-D-A. Is that considered the same? No, because the problem says two seatings are the same if one is a rotation of the other. So, if we consider rotations as the same, then A-B-C-D is the same as B-C-D-A, etc.

Therefore, if we fix A's position, we are effectively counting each unique arrangement once.

But if we don't fix positions, how many unique arrangements are there?

Each unique arrangement can be rotated so that A is in a specific position. Therefore, the number of unique arrangements is equal to the number of arrangements with A fixed.

Therefore, if fixing A's position gives 4 arrangements, then the total number of unique circular arrangements is 4.

But according to the formula (number of linear arrangements)/n, which was 12 / 4 = 3.

So, there's a contradiction. Therefore, my formula is wrong.

So, what is the correct formula?

Perhaps the formula is different. Maybe when considering circular arrangements with blocks, we need to use (k-1)! * internal arrangements, but in the case where the block is part of the total.

Wait, in the case of 4 people with A and B together, treating them as a block, we have 3 entities: [AB], C, D.

Number of circular arrangements of these 3 entities is (3-1)! = 2. Then, multiply by internal arrangements of the block: 2! = 2. So total is 2 * 2 = 4, which matches the manual count.

Ah! So, in this case, the formula (k-1)! * product of internal arrangements works.

Earlier, when I tried with k=1, it didn't work, but maybe that's a special case?

Wait, let's check with k=1. If we have 3 people as a single block, then k=1. So, (1-1)! * internal arrangements = 1 * 3! = 6. But the actual number of circular arrangements is 2.

But in the case of k=3, with 3 entities, it works.

Wait, so why is there a discrepancy?

Wait, maybe when the block is the entire set, the formula is different?

Wait, if we have 3 people as a single block, then the number of circular arrangements is (number of circular arrangements of the block). But the block is 3 people, so the number of circular arrangements is (3-1)! = 2. But according to the formula (k-1)! * internal arrangements, where k=1, it's 0! * 3! = 6.

So, clearly, the formula is not applicable when the block is the entire set.

But in the case of 4 people with a block of 2, the formula works.

So, what's the difference?

Ah! Maybe when the block is not the entire set, the formula (k-1)! * product of internal arrangements is correct. But when the block is the entire set, it's different.

But why?

Wait, in the case of the block being the entire set, the number of circular arrangements is (n-1)! where n is the total number of people. But according to the formula (k-1)! * product of internal arrangements, where k=1, it's 0! * n! = n! which is different.

Therefore, the formula works when there are multiple blocks, but not when there is only one block.

But in the case of 4 people with a block of 2, we have k=3 entities (block, C, D). So, k=3, and the formula (k-1)! * internal arrangements = 2! * 2! = 4, which matches the manual count.

So, maybe the formula is correct when there are multiple blocks.

Therefore, going back to the original problem.

Original problem: 8 people, with 3 people (Pierre, Rosa, Thomas) sitting together.

So, we have 1 block (P-R-T) and 5 other people. So, total of 6 entities.

Therefore, number of circular arrangements is (6-1)! * internal arrangements of the block = 5! * 3! = 120 * 6 = 720.

But earlier, when I tried with the formula (number of linear arrangements)/n, I got 540.

But in the case of 4 people with a block of 2, the formula (k-1)! * internal arrangements worked.

So, why the discrepancy?

Wait, in the case of 4 people with a block of 2, the number of circular arrangements is (k-1)! * internal arrangements = 2! * 2! = 4, which matches the manual count.

But if I use the formula (number of linear arrangements)/n, where number of linear arrangements is 3! * 2! = 12, divided by n=4, gives 3, which is wrong.

Therefore, the formula (k-1)! * internal arrangements is correct, and the other formula is wrong.

So, why did the formula (number of linear arrangements)/n give a wrong answer?

Because when we treat the block as an entity, the number of linear arrangements is k! * product of internal arrangements, where k is the number of entities. But for circular arrangements, it's (k-1)! * product of internal arrangements.

Therefore, the correct formula is (k-1)! * product of internal arrangements.

But in the case of k=1, this formula gives (1-1)! * internal arrangements = 1 * n! which is wrong. But in reality, when k=1, the number of circular arrangements is (n-1)!

So, there's an inconsistency here.

But in the case where we have multiple blocks, the formula works.

Therefore, maybe the formula is correct when there are multiple blocks, and when there is only one block, we need to adjust.

But in our original problem, we have multiple blocks (the block of 3 and 5 other individuals). Therefore, the formula (k-1)! * product of internal arrangements is correct.

Therefore, going back to the original problem:

Number of circular arrangements = (6-1)! * 3! = 5! * 6 = 120 * 6 = 720.

But earlier, when I thought of fixing the block, I thought of fixing the block's position to eliminate rotations.

Alternatively, another way to think about it:

In circular permutations, fixing one position is a common technique. So, if we fix the block of Pierre, Rosa, and Thomas in a specific position, then we can arrange the remaining 5 people in the remaining seats.

But since the block is fixed, we don't have to worry about rotations.

However, within the block, Pierre, Rosa, and Thomas can be arranged in 3! ways.

The remaining 5 people can be arranged in 5! ways.

Therefore, total number of arrangements is 3! * 5! = 6 * 120 = 720.

Which matches the previous result.

But why did the other approach give a different answer?

Because when I considered the formula (number of linear arrangements)/n, I think I was making a mistake in how I calculated the linear arrangements.

Wait, if we fix the block's position, then the number of linear arrangements is 5! * 3! = 720. But since we fixed the block's position, we don't need to divide by anything.

But in circular arrangements, fixing one position is equivalent to considering the number of circular arrangements as (number of linear arrangements with one position fixed).

Therefore, the answer is 5! * 3! = 720.

Therefore, the correct answer is 720.

But then why did the previous example with 4 people and a block of 2 give a different result when I tried the formula (number of linear arrangements)/n?

Because in that case, if I fix the block's position, then the number of arrangements is 2! * 2! = 4, which matches the manual count.

But if I use the formula (number of linear arrangements)/n, where number of linear arrangements is 3! * 2! = 12, divided by n=4, gives 3, which is wrong.

Therefore, the formula (number of linear arrangements)/n is not correct.

Therefore, the correct approach is to treat the block as an entity, calculate (k-1)! * product of internal arrangements, where k is the number of entities.

Therefore, in the original problem, the answer is 5! * 3! = 720.

But then why did the example with 3 people give a discrepancy?

Because when k=1, the formula gives (1-1)! * internal arrangements = 1 * 3! = 6, but the actual number of circular arrangements is 2.

But in that case, the block is the entire set, so maybe the formula is not applicable?

But in our original problem, the block is not the entire set.

Therefore, the formula works when there are multiple blocks.

Therefore, the answer is 720.

But let me check with another example.

Suppose we have 5 people, and 2 of them must sit together.

Number of circular arrangements:

Treat the 2 as a block. So, k=4 entities (block + 3 others).

Number of circular arrangements: (4-1)! * 2! = 6 * 2 = 12.

Alternatively, fix the block's position. Then, the remaining 3 people can be arranged in 3! ways, and the block can be arranged in 2! ways. Total: 3! * 2! = 12.

Which is correct.

Alternatively, if I use the formula (number of linear arrangements)/n:

Number of linear arrangements is 4! * 2! = 24 * 2 = 48. Divide by 5 gives 9.6, which is not an integer. So, clearly wrong.

Therefore, the formula (number of linear arrangements)/n is not correct.

Therefore, the correct formula is (k-1)! * product of internal arrangements.

Therefore, returning to the original problem, the answer is 5! * 3! = 720.

But let me check with another approach.

Imagine that we have 8 seats around a table. We want Pierre, Rosa, and Thomas to sit together.

First, consider the number of ways to seat them together.

First, we can think of the number of ways to choose a block of 3 consecutive seats. In a circular table, the number of ways to choose a block of 3 consecutive seats is 8 (since there are 8 seats, each seat can be the start of a block of 3).

But since rotations are considered the same, we need to be careful.

Wait, but if we fix the block's position, then we can avoid overcounting.

Alternatively, if we don't fix positions, the number of ways to arrange the block is 8 (positions for the block) * 3! (arrangements within the block) * 5! (arrangements of the others). But since rotations are considered the same, we have overcounted by a factor of 8.

Therefore, total number of arrangements is (8 * 3! * 5!) / 8 = 3! * 5! = 720.

Ah! So this approach gives the same result as before.

Therefore, this confirms that the answer is 720.

Therefore, the initial formula (k-1)! * product of internal arrangements is correct, and the confusion with the single block case is because when the block is the entire set, the formula doesn't apply, but in our problem, we have multiple blocks.

Therefore, the answer is 5! * 3! = 720.

But let me check with the example of 3 people. If we have 3 people, and we want them to sit together.

Number of circular arrangements:

Using this approach: number of ways to choose a block of 3 seats: 3 (since circular table, number of blocks of 3 consecutive seats is 3). Then, arrangements: 3 * 3! / 3 = 3! = 6. But the actual number of circular arrangements is 2.

Wait, discrepancy again.

But why?

Because when we have the entire set as a block, the number of ways to choose the block is equal to the number of seats, but since rotations are considered the same, we have overcounted.

Wait, in this case, if we have 3 people, and we consider the number of ways to seat them together:

Number of ways to choose the block: 3 (positions). Then, arrangements within the block: 3!. Then, divide by 3 (number of rotations) to get 3! / 3 = 2, which is correct.

Ah! So, in this case, the formula is (number of ways to choose the block * internal arrangements) / n.

So, in general, the formula is:

Number of circular arrangements = (number of ways to choose the block * internal arrangements of the block * arrangements of the rest) / n.

But in our original problem, the number of ways to choose the block is 8 (since there are 8 seats, and each block of 3 consecutive seats can be placed in 8 different positions).

But since rotations are considered the same, we need to divide by n=8.

Therefore, total number of arrangements is (8 * 3! * 5!) / 8 = 3! * 5! = 720.

Which matches the previous result.

Therefore, this formula works.

So, in the case of 3 people:

Number of ways to choose the block: 3.

Internal arrangements: 3!.

Arrangements of the rest: since there are no others, it's 1.

Total: (3 * 3! * 1) / 3 = 3! = 6. But the actual number of circular arrangements is 2.

Wait, discrepancy again.

Wait, what's wrong here?

Ah! Because when there are no other people, the arrangements of the rest is 1, but we also need to consider that the block is the entire set.

Wait, but if we have 3 people, and we want to arrange them around a table with the block of 3 people, then the number of circular arrangements is (number of ways to choose the block * internal arrangements) / n.

But number of ways to choose the block is 3, internal arrangements is 3!, divided by n=3 gives 3! = 6. But the actual number is 2.

So, why is this?

Because when we choose the block, we are considering different positions, but in circular arrangements, those are considered the same.

Wait, but if we have 3 people, and we fix the block's position, then the number of arrangements is 3! / 3 = 2.

But according to the formula (number of ways to choose the block * internal arrangements) / n, it's (3 * 3!) / 3 = 6.

So, why is this different?

Because when we choose the block, we are considering different positions, but in reality, those positions are rotations of each other.

Therefore, the formula (number of ways to choose the block * internal arrangements) / n is not correct.

But in the case of the original problem, where there are other people, the formula works.

Wait, in the original problem, when we have other people, the arrangements of the rest are different for each block position.

Wait, no. If we have other people, then choosing different positions for the block would lead to different arrangements.

But in the case of 3 people, there are no other people, so choosing different positions for the block doesn't change the arrangement.

Therefore, the formula (number of ways to choose the block * internal arrangements) / n is not correct in general.

But in the case where there are other people, it works?

Wait, in the case of 4 people with a block of 2, using this formula:

Number of ways to choose the block: 4 (since there are 4 seats, each block of 2 consecutive seats can be placed in 4 positions).

Internal arrangements: 2!.

Arrangements of the rest: 2! (since there are 2 other people).

Total: (4 * 2! * 2!) / 4 = 2! * 2! = 4, which matches the manual count.

So, in this case, it works.

But in the case of 3 people, it doesn't.

So, why?

Because when there are other people, the arrangements of the rest are different for each block position, so dividing by n accounts for the overcounting.

But when there are no other people, the arrangements of the rest are 1, and dividing by n gives an incorrect result.

Therefore, the formula works when there are other people, but not when the block is the entire set.

Therefore, in our original problem, since there are other people, the formula works.

Therefore, the answer is 720.

Therefore, after all this thinking, I believe the correct answer is 720.

**Final Answer**
The number of ways is \boxed{720}.

To determine the number of ways 8 people can sit around a round table such that **Pierre, Rosa, and Thomas all sit together**, we need to carefully account for both the **circular nature of the table** and the **restriction that three specific people must be adjacent**.

---

### Step 1: Treat the Group as a Single Block

Since **Pierre, Rosa, and Thomas** must sit together, we can treat them as a **single block** or unit. This reduces the problem from arranging 8 individuals to arranging **6 entities**: the block of 3 people and the other 5 individuals.

---

### Step 2: Circular Permutations of the 6 Entities

In a **circular arrangement**, the number of distinct ways to arrange $ n $ entities is:

$$
(n - 1)!
$$

So, for 6 entities (the block and the 5 others), the number of circular arrangements is:

$$
(6 - 1)! = 5! = 120
$$

---

### Step 3: Internal Arrangements Within the Block

Within the block of 3 people (Pierre, Rosa, and Thomas), they can be arranged among themselves in:

$$
3! = 6
$$

ways.

---

### Step 4: Total Number of Arrangements

To find the total number of valid arrangements, we multiply the number of circular arrangements of the 6 entities by the number of internal arrangements within the block:

$$
5! \times 3! = 120 \times 6 = 720
$$

---

### Final Answer

$$
\boxed{720}
$$

Solution: First choose three consecutive seats for Pierre, Rosa, and Thomas.  It doesn't matter which three consecutive seats that we choose, since any three such seats can be rotated to any other such seats.  Once the three seats are chosen, there are $3!$ ways to seat the three friends there.  The other five seats are for the other five people, so there are $5!$ ways to seat them there.  The answer is $3! \times 5! = \boxed{720}$.


Explanation:
Okay, so I need to find the eighth term of this geometric sequence: 125/9, 25/3, 5, 3, ... Let me think about how to approach this.

First, I remember that in a geometric sequence, each term is obtained by multiplying the previous term by a common ratio, right? So, the key here is to figure out what the common ratio (r) is. Once I have that, I can use the formula for the nth term of a geometric sequence to find the eighth term.

Let me start by finding the common ratio. To do that, I can divide the second term by the first term. Let me write that down:

First term (a₁) = 125/9

Second term (a₂) = 25/3

So, the common ratio r = a₂ / a₁ = (25/3) / (125/9). Hmm, dividing fractions... I need to multiply by the reciprocal. So that would be (25/3) * (9/125). Let me compute that.

First, simplify the numbers. 25 and 125 have a common factor. 25 divides into 125 five times. So, 25/125 is 1/5. Then, 9/3 is 3. So, multiplying those together: (1/5) * 3 = 3/5. Wait, is that right? Let me check again.

Wait, (25/3) divided by (125/9) is equal to (25/3) * (9/125). Let me compute numerator and denominator separately. Numerator: 25 * 9 = 225. Denominator: 3 * 125 = 375. So, 225/375. Simplify that. Both divide by 75? 225 ÷ 75 = 3, 375 ÷ 75 = 5. So, 3/5. Yes, same result. So the common ratio is 3/5.

Let me check with the next terms to make sure. The third term is 5. Let me see if multiplying the second term by 3/5 gives me 5.

Second term is 25/3. Multiply by 3/5: (25/3)*(3/5). The 3s cancel out, so 25/5 = 5. Perfect, that's the third term. Then the fourth term is 3. Let me check that. Multiply third term (5) by 3/5: 5*(3/5) = 3. Yep, that works. So the common ratio is indeed 3/5.

Okay, so now that I know the first term and the common ratio, I can use the formula for the nth term of a geometric sequence. The formula is:

aₙ = a₁ * r^(n-1)

So, for the eighth term, n = 8. Let me plug in the values:

a₈ = (125/9) * (3/5)^(8-1) = (125/9) * (3/5)^7

Now, I need to compute (3/5)^7 and then multiply by 125/9. Let me compute that step by step.

First, let me compute (3/5)^7. Let me write that as 3^7 / 5^7. Let me calculate 3^7 and 5^7.

Starting with 3^7:

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

3^7 = 2187

Okay, so 3^7 is 2187.

Now 5^7:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

5^6 = 15625

5^7 = 78125

So, (3/5)^7 = 2187 / 78125.

Therefore, a₈ = (125/9) * (2187 / 78125). Let me compute this multiplication.

First, let me write it as (125 * 2187) / (9 * 78125). Let me see if I can simplify this fraction before multiplying everything out.

Looking at the numerator and denominator, I can look for common factors. Let me note that 125 is 5^3, and 78125 is 5^7. Also, 2187 is 3^7, and 9 is 3^2.

So, let me write the numerator and denominator in terms of prime factors:

Numerator: 125 * 2187 = 5^3 * 3^7

Denominator: 9 * 78125 = 3^2 * 5^7

So, the fraction becomes (5^3 * 3^7) / (3^2 * 5^7) = (3^(7-2)) / (5^(7-3)) = 3^5 / 5^4.

Wait, that's a much simpler way to do it! Instead of multiplying out the large numbers, I can cancel out the common factors. Let me verify that.

Yes, because when you divide exponents with the same base, you subtract the exponents. So, for the 3s: 3^7 divided by 3^2 is 3^(7-2) = 3^5. For the 5s: 5^3 divided by 5^7 is 5^(3-7) = 5^(-4) = 1/(5^4). Wait, but in the original fraction, the denominator is 5^7 and the numerator is 5^3, so it's 5^3 / 5^7 = 1 / 5^4. So overall, the entire fraction is 3^5 / 5^4.

So, that simplifies the calculation. Let me compute 3^5 and 5^4.

3^5 is 243 (since 3^4 is 81, times 3 is 243).

5^4 is 625 (since 5^2 is 25, squared is 625).

Therefore, the eighth term is 243 / 625.

Wait, let me check if that's correct. Let me verify by computing the original multiplication:

(125/9) * (2187 / 78125). Let me compute numerator and denominator separately.

Numerator: 125 * 2187. Let me compute that.

Well, 125 * 2187. Since 125 is 1000/8, so 2187 / 8 * 1000. Wait, maybe that's not helpful. Alternatively, note that 2187 divided by 9 is 243. Because 9*243 = 2187. Let me check: 243*9. 200*9=1800, 40*9=360, 3*9=27. 1800+360=2160+27=2187. Yes! So 2187 / 9 = 243. Therefore, the numerator is 125 * 2187 = 125 * 9 * 243. Wait, but the denominator is 9 * 78125. So, if I write the entire fraction:

(125 * 2187) / (9 * 78125) = (125 / 78125) * (2187 / 9) = (125 / 78125) * 243.

Now, 125 / 78125. Let me compute that. 78125 divided by 125. Let me see, 125 * 625 = 78125, because 125 * 500 = 62,500 and 125 * 125 = 15,625. So 62,500 + 15,625 = 78,125. Therefore, 125 / 78125 = 1 / 625.

Therefore, the entire fraction becomes (1 / 625) * 243 = 243 / 625.

So that's the same result as before. So, 243/625.

Let me check if this fraction can be simplified further. Let's see, 243 and 625.

243 is 3^5, and 625 is 5^4. They have no common factors besides 1, so the fraction is already in its simplest form.

Therefore, the eighth term is 243/625.

But just to be thorough, let me try another approach. Let me compute each term step by step up to the eighth term, to verify.

First term: 125/9 ≈ 13.888...

Second term: 125/9 * 3/5 = (125*3)/(9*5) = 375/45 = 25/3 ≈ 8.333... Correct.

Third term: 25/3 * 3/5 = (25*3)/(3*5) = 75/15 = 5. Correct.

Fourth term: 5 * 3/5 = 3. Correct.

Fifth term: 3 * 3/5 = 9/5 = 1.8

Sixth term: 9/5 * 3/5 = 27/25 = 1.08

Seventh term: 27/25 * 3/5 = 81/125 = 0.648

Eighth term: 81/125 * 3/5 = 243/625.

Yes! So that's the same result. So, the eighth term is 243/625.

Therefore, I can be confident that the answer is 243/625.

But just to make sure I didn't make any calculation errors in the exponents or simplifications, let me recap:

Original formula: a₈ = a₁ * r^(7). Since n=8, exponent is 7.

a₁ = 125/9, r = 3/5.

So, 125/9 * (3/5)^7.

Breaking down (3/5)^7: 3^7 / 5^7.

Multiply by 125/9: (125 * 3^7) / (9 * 5^7).

Note that 125 is 5^3, and 9 is 3^2.

So, substituting: (5^3 * 3^7) / (3^2 * 5^7) = 3^(7-2) / 5^(7-3) = 3^5 / 5^4 = 243 / 625.

Yep, same result.

Alternatively, if I think about the terms:

Term 1: 125/9

Term 2: 125/9 * 3/5 = 125/9 * 3/5 = (125/5) * (3/9) = 25 * 1/3 = 25/3. Correct.

Term 3: 25/3 * 3/5 = 25/5 = 5. Correct.

Term 4: 5 * 3/5 = 3. Correct.

Term 5: 3 * 3/5 = 9/5.

Term 6: 9/5 * 3/5 = 27/25.

Term 7: 27/25 * 3/5 = 81/125.

Term 8: 81/125 * 3/5 = 243/625.

Yep, same answer.

So, I think that's solid. Therefore, the eighth term is 243/625.

**Final Answer**
The eighth term of the sequence is \boxed{\dfrac{243}{625}}.

To find the eighth term of the geometric sequence:

$$
\frac{125}{9}, \frac{25}{3}, 5, 3, \ldots
$$

we begin by identifying the **common ratio** $ r $, which is the factor by which each term is multiplied to get the next term.

---

### Step 1: Find the Common Ratio

We compute the ratio between the second and first terms:

$$
r = \frac{a_2}{a_1} = \frac{\frac{25}{3}}{\frac{125}{9}} = \frac{25}{3} \cdot \frac{9}{125} = \frac{225}{375} = \frac{3}{5}
$$

We can verify this ratio with the next terms:

- $ a_3 = a_2 \cdot r = \frac{25}{3} \cdot \frac{3}{5} = 5 $
- $ a_4 = a_3 \cdot r = 5 \cdot \frac{3}{5} = 3 $

So the common ratio is confirmed to be $ r = \frac{3}{5} $.

---

### Step 2: Use the Formula for the $ n $-th Term of a Geometric Sequence

The general formula for the $ n $-th term of a geometric sequence is:

$$
a_n = a_1 \cdot r^{n-1}
$$

We are asked to find the **eighth term**, so $ n = 8 $, $ a_1 = \frac{125}{9} $, and $ r = \frac{3}{5} $. Plugging in:

$$
a_8 = \frac{125}{9} \cdot \left( \frac{3}{5} \right)^7
$$

---

### Step 3: Simplify the Expression

We simplify $ \left( \frac{3}{5} \right)^7 $ as:

$$
\left( \frac{3}{5} \right)^7 = \frac{3^7}{5^7} = \frac{2187}{78125}
$$

Now multiply:

$$
a_8 = \frac{125}{9} \cdot \frac{2187}{78125}
$$

We simplify the expression by factoring:

- $ 125 = 5^3 $
- $ 78125 = 5^7 $
- $ 2187 = 3^7 $
- $ 9 = 3^2 $

So:

$$
a_8 = \frac{5^3 \cdot 3^7}{3^2 \cdot 5^7} = \frac{3^{7-2}}{5^{7-3}} = \frac{3^5}{5^4}
$$

Now compute the powers:

- $ 3^5 = 243 $
- $ 5^4 = 625 $

Thus:

$$
a_8 = \frac{243}{625}
$$

---

### Final Answer

$$
\boxed{\dfrac{243}{625}}
$$

[Assistant]
Explanation:
Okay, so I need to find the eighth term of this geometric sequence: 125/9, 25/3, 5, 3, ... Let me think about how to approach this.

First, I remember that in a geometric sequence, each term is obtained by multiplying the previous term by a common ratio, right? So, the key here is to figure out what the common ratio (r) is. Once I have that, I can use the formula for the nth term of a geometric sequence to find the eighth term.

Let me start by finding the common ratio. To do that, I can divide the second term by the first term. Let me write that down:

First term (a₁) = 125/9

Second term (a₂) = 25/3

So, the common ratio r = a₂ / a₁ = (25/3) / (125/9). Hmm, dividing fractions... I need to multiply by the reciprocal. So that would be (25/3) * (9/125). Let me compute that.

First, simplify the numbers. 25 and 125 have a common factor. 25 divides into 125 five times. So, 25/125 is 1/5. Then, 9/3 is 3. So, multiplying those together: (1/5) * 3 = 3/5. Wait, is that right? Let me check again.

Wait, (25/3) divided by (125/9) is equal to (25/3) * (9/125). Let me compute numerator and denominator separately. Numerator: 25 * 9 = 225. Denominator: 3 * 125 = 375. So, 225/375. Simplify that. Both divide by 75? 225 ÷ 75 = 3, 375 ÷ 75 = 5. So, 3/5. Yes, same result. So the common ratio is 3/5.

Let me check with the next terms to make sure. The third term is 5. Let me see if multiplying the second term by 3/5 gives me 5.

Second term is 25/3. Multiply by 3/5: (25/3)*(3/5). The 3s cancel out, so 25/5 = 5. Perfect, that's the third term. Then the fourth term is 3. Let me check that. Multiply third term (5) by 3/5: 5*(3/5) = 3. Yep, that works. So the common ratio is indeed 3/5.

Okay, so now that I know the first term and the common ratio, I can use the formula for the nth term of a geometric sequence. The formula is:

aₙ = a₁ * r^(n-1)

So, for the eighth term, n = 8. Let me plug in the values:

a₈ = (125/9) * (3/5)^(8-1) = (125/9) * (3/5)^7

Now, I need to compute (3/5)^7 and then multiply by 125/9. Let me compute that step by step.

First, let me compute (3/5)^7. Let me write that as 3^7 / 5^7. Let me calculate 3^7 and 5^7.

Starting with 3^7:

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

3^7 = 2187

Okay, so 3^7 is 2187.

Now 5^7:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

5^6 = 15625

5^7 = 78125

So, (3/5)^7 = 2187 / 78125.

Therefore, a₈ = (125/9) * (2187 / 78125). Let me compute this multiplication.

First, let me write it as (125 * 2187) / (9 * 78125). Let me see if I can simplify this fraction before multiplying everything out.

Looking at the numerator and denominator, I can look for common factors. Let me note that 125 is 5^3, and 78125 is 5^7. Also, 2187 is 3^7, and 9 is 3^2.

So, let me write the numerator and denominator in terms of prime factors:

Numerator: 125 * 2187 = 5^3 * 3^7

Denominator: 9 * 78125 = 3^2 * 5^7

So, the fraction becomes (5^3 * 3^7) / (3^2 * 5^7) = (3^(7-2)) / (5^(7-3)) = 3^5 / 5^4.

Wait, that's a much simpler way to do it! Instead of multiplying out the large numbers, I can cancel out the common factors. Let me verify that.

Yes, because when you divide exponents with the same base, you subtract the exponents. So, for the 3s: 3^7 divided by 3^2 is 3^(7-2) = 3^5. For the 5s: 5^3 divided by 5^7 is 5^(3-7) = 5^(-4) = 1/(5^4). Wait, but in the original fraction, the denominator is 5^7 and the numerator is 5^3, so it's 5^3 / 5^7 = 1 / 5^4. So overall, the entire fraction is 3^5 / 5^4.

So, that simplifies the calculation. Let me compute 3^5 and 5^4.

3^5 is 243 (since 3^4 is 81, times 3 is 243).

5^4 is 625 (since 5^2 is 25, squared is 625).

Therefore, the eighth term is 243 / 625.

Wait, let me check if that's correct. Let me verify by computing the original multiplication:

(125/9) * (2187 / 78125). Let me compute numerator and denominator separately.

Numerator: 125 * 2187. Let me compute that.

Well, 125 * 2187. Since 125 is 1000/8, so 2187 / 8 * 1000. Wait, maybe that's not helpful. Alternatively, note that 2187 divided by 9 is 243. Because 9*243 = 2187. Let me check: 243*9. 200*9=1800, 40*9=360, 3*9=27. 1800+360=2160+27=2187. Yes! So 2187 / 9 = 243. Therefore, the numerator is 125 * 2187 = 125 * 9 * 243. Wait, but the denominator is 9 * 78125. So, if I write the entire fraction:

(125 * 2187) / (9 * 78125) = (125 / 78125) * (2187 / 9) = (125 / 78125) * 243.

Now, 125 / 78125. Let me compute that. 78125 divided by 125. Let me see, 125 * 625 = 78125, because 125 * 500 = 62,500 and 125 * 125 = 15,625. So 62,500 + 15,625 = 78,125. Therefore, 125 / 78125 = 1 / 625.

Therefore, the entire fraction becomes (1 / 625) * 243 = 243 / 625.

So that's the same result as before. So, 243/625.

Let me check if this fraction can be simplified further. Let's see, 243 and 625.

243 is 3^5, and 625 is 5^4. They have no common factors besides 1, so the fraction is already in its simplest form.

Therefore, the eighth term is 243/625.

But just to be thorough, let me try another approach. Let me compute each term step by step up to the eighth term, to verify.

First term: 125/9 ≈ 13.888...

Second term: 125/9 * 3/5 = (125*3)/(9*5) = 375/45 = 25/3 ≈ 8.333... Correct.

Third term: 25/3 * 3/5 = (25*3)/(3*5) = 75/15 = 5. Correct.

Fourth term: 5 * 3/5 = 3. Correct.

Fifth term: 3 * 3/5 = 9/5 = 1.8

Sixth term: 9/5 * 3/5 = 27/25 = 1.08

Seventh term: 27/25 * 3/5 = 81/125 = 0.648

Eighth term: 81/125 * 3/5 = 243/625.

Yes! So that's the same result. So, the eighth term is 243/625.

Therefore, I can be confident that the answer is 243/625.

But just to make sure I didn't make any calculation errors in the exponents or simplifications, let me recap:

Original formula: a₈ = a₁ * r^(7). Since n=8, exponent is 7.

a₁ = 125/9, r = 3/5.

So, 125/9 * (3/5)^7.

Breaking down (3/5)^7: 3^7 / 5^7.

Multiply by 125/9: (125 * 3^7) / (9 * 5^7).

Note that 125 is 5^3, and 9 is 3^2.

So, substituting: (5^3 * 3^7) / (3^2 * 5^7) = 3^(7-2) / 5^(7-3) = 3^5 / 5^4 = 243 / 625.

Yep, same result.

Alternatively, if I think about the terms:

Term 1: 125/9

Term 2: 125/9 * 3/5 = 125/9 * 3/5 = (125/5) * (3/9) = 25 * 1/3 = 25/3. Correct.

Term 3: 25/3 * 3/5 = 25/5 = 5. Correct.

Term 4: 5 * 3/5 = 3. Correct.

Term 5: 3 * 3/5 = 9/5.

Term 6: 9/5 * 3/5 = 27/25.

Term 7: 27/25 * 3/5 = 81/125.

Term 8: 81/125 * 3/5 = 243/625.

Yep, same answer.

So, I think that's solid. Therefore, the eighth term is 243/625.

**Final Answer**
The eighth term of the sequence is \boxed{\dfrac{243}{625}}.

To find the eighth term of the geometric sequence:

$$
\frac{125}{9}, \frac{25}{3}, 5, 3, \ldots
$$

we begin by identifying the **common ratio** $ r $, which is the factor by which each term is multiplied to get the next term.

---

### Step 1: Find the Common Ratio

We compute the ratio between the second and first terms:

$$
r = \frac{a_2}{a_1} = \frac{\frac{25}{3}}{\frac{125}{9}} = \frac{25}{3} \cdot \frac{9}{125} = \frac{225}{375} = \frac{3}{5}
$$

We can verify this ratio with the next terms:

- $ a_3 = a_2 \cdot r = \frac{25}{3} \cdot \frac{3}{5} = 5 $
- $ a_4 = a_3 \cdot r = 5 \cdot \frac{3}{5} = 3 $

So the common ratio is confirmed to be $ r = \frac{3}{5} $.

---

### Step 2: Use the Formula for the $ n $-th Term of a Geometric Sequence

The general formula for the $ n $-th term of a geometric sequence is:

$$
a_n = a_1 \cdot r^{n-1}
$$

We are asked to find the **eighth term**, so $ n = 8 $, $ a_1 = \frac{125}{9} $, and $ r = \frac{3}{5} $. Plugging in:

$$
a_8 = \frac{125}{9} \cdot \left( \frac{3}{5} \right)^7
$$

---

### Step 3: Simplify the Expression

We simplify $ \left( \frac{3}{5} \right)^7 $ as:

$$
\left( \frac{3}{5} \right)^7 = \frac{3^7}{5^7} = \frac{2187}{78125}
$$

Now multiply:

$$
a_8 = \frac{125}{9} \cdot \frac{2187}{78125}
$$

We simplify the expression by factoring:

- $ 125 = 5^3 $
- $ 78125 = 5^7 $
- $ 2187 = 3^7 $
- $ 9 = 3^2 $

So:

$$
a_8 = \frac{5^3 \cdot 3^7}{3^2 \cdot 5^7} = \frac{3^{7-2}}{5^{7-3}} = \frac{3^5}{5^4}
$$

Now compute the powers:

- $ 3^5 = 243 $
- $ 5^4 = 625 $

Thus:

$$
a_8 = \frac{243}{625}
$$

---

### Final Answer

$$
\boxed{\dfrac{243}{625}}
$$

Solution:
The common ratio between consecutive terms is $\frac{3}{5}$ (you can choose any two consecutive terms and divide the second one by the first to find the common ratio). So the $n^\text{th}$ term of the sequence is $\frac{125}{9} \cdot \left( \frac{3}{5} \right)^{n-1}$. Plugging in $n=8$, we get $$
\frac{125}{9} \cdot \left( \frac{3}{5} \right)^{7} = \frac{5^3}{3^2} \cdot \frac{3^7}{5^7}
= \frac{3^5}{5^4}
= \boxed{\frac{243}{625}}.
$$