Explanation:
Okay, so I need to figure out how many ways there are to put 5 indistinguishable balls into 2 indistinguishable boxes. Hmm, let me think.

First, I remember that when dealing with problems involving distributing objects into boxes, the key factors are whether the objects are distinguishable and whether the boxes are distinguishable. In this case, both the balls and the boxes are not distinguishable. That probably means that the order of the boxes doesn't matter, and the balls are all the same.

Let me recall some combinatorial concepts. If the boxes were distinguishable, like Box A and Box B, then the problem would be simpler. For each ball, you could choose which box it goes into. Since the balls are indistinct, the number of ways would be the number of non-negative integer solutions to the equation:

b1 + b2 = 5

where b1 is the number of balls in box 1 and b2 is the number in box 2. The formula for that is C(n + k - 1, k - 1), where n is the number of balls and k is the number of boxes. So here, that would be C(5 + 2 - 1, 2 - 1) = C(6,1) = 6. But wait, that's if the boxes are distinguishable. However, in our problem, the boxes are not distinguishable. So some of these distributions might be counted multiple times in the distinguishable case, and we need to adjust for that.

Let me think. If the boxes are indistinct, then putting 1 ball in box A and 4 in box B is the same as putting 1 in box B and 4 in box A. So we need to count the number of distinct distributions where the order doesn't matter.

So, for the case of indistinct boxes, the number of ways is equal to the number of integer partitions of 5 into at most 2 parts. Since we have two boxes, each partition will have two parts (which can be equal or not).

Let me list all the possible partitions of 5 into at most 2 parts:

1. 5 + 0
2. 4 + 1
3. 3 + 2

Wait, that's three partitions. But hold on, if we consider that the boxes are indistinct, then partitions like 0 + 5 are the same as 5 + 0, so we don't count them separately. Similarly, 1 + 4 is same as 4 + 1, and 2 + 3 is same as 3 + 2. So the distinct distributions are just those three.

But wait, is that all? Let me check. For example, if we have two boxes, the number of ways to distribute 5 indistinct balls is equal to the floor of (n + 1)/2? Wait, no. Wait, for n=5, the number of partitions into at most 2 parts is equal to the number of integer partitions of 5 where the number of parts is 1 or 2.

But since the boxes are indistinct, we can have:

- All balls in one box: that's 1 way (since both boxes are the same, putting all in one is the same regardless of which box you choose).

- Splitting into two boxes: the number of ways is equal to the number of ways to split 5 into two positive integers where order doesn't matter. Which would be floor(5/2) = 2? Wait, let me think again.

For splitting into two positive integers, the number of partitions is equal to the floor of n/2. Wait, for n=5, the splits are 1+4 and 2+3. So that's two splits. So total number of ways would be 1 (all in one box) + 2 (split into two boxes) = 3.

So that would be 3 ways. Therefore, the answer is 3? But let me verify this with another approach to make sure.

Alternatively, since the boxes are indistinct, the problem reduces to finding the number of equivalence classes under the action of swapping boxes. In the case of distinguishable boxes, we have 6 distributions. But when boxes are indistinct, each distribution is counted twice except when the distribution is symmetric (i.e., when both boxes have the same number of balls).

So, in the case of 5 balls and 2 boxes, how many symmetric distributions are there? Well, for the number of balls in each box to be equal, we need 5 to be even, but 5 is odd. Therefore, there are no symmetric distributions. Therefore, all distributions are counted twice in the distinguishable case. Therefore, the number of distinct distributions is 6 / 2 = 3.

Yes, that matches the previous result. Therefore, the answer is 3.

But let me think again. Let me list all possibilities.

If the boxes are indistinct, then the possible distributions are:

1. All 5 balls in one box, and 0 in the other. Since boxes are indistinct, there's only one way to do this.

2. 4 balls in one box, 1 in the other. Again, since boxes are indistinct, this is only one way.

3. 3 balls in one box, 2 in the other. Again, only one way.

So that's three different ways.

Alternatively, if I think of the problem as the number of ways to partition the integer 5 into up to 2 parts, which is exactly what I did before.

But let me check with another method. Suppose I use the formula for the number of ways to distribute n indistinct objects into k indistinct boxes, which is equal to the number of integer partitions of n into at most k parts.

In this case, n=5, k=2. The number of integer partitions of 5 into at most 2 parts is indeed 3, as listed above.

Alternatively, if I recall that the number of ways is equal to the floor(n/2) + 1? Wait, for n=5, floor(5/2) is 2, plus 1 is 3. That works. For example, if n=4, floor(4/2) +1 = 2 +1=3. Let's check: partitions of 4 into at most 2 parts: 4+0, 3+1, 2+2. That's 3, which matches. So yes, the formula seems to be floor(n/2) +1? Wait, but for n=3, floor(3/2) +1=1+1=2. Partitions of 3 into at most 2 parts: 3+0, 2+1. That's 2, which matches. For n=2: floor(2/2)+1=1+1=2. Partitions: 2+0, 1+1. Correct. For n=1: floor(1/2)+1=0+1=1. Partition: 1+0. Correct. So yes, the formula seems to be floor(n/2) +1? Wait, but for n=5, floor(5/2)=2, so 2+1=3. So that works.

But wait, actually, the formula for the number of partitions of n into at most k parts is not exactly floor(n/2) +1. Wait, that formula seems to be specific for k=2. For general k, it's more complicated. But in our case, since k=2, the number of partitions of n into at most 2 parts is equal to the number of integer partitions of n where the largest part is at most 2. Wait, no, actually, the number of partitions of n into at most 2 parts is equal to the number of partitions of n where the number of parts is at most 2. Which is equivalent to the number of partitions of n into 1 or 2 parts.

But for the number of partitions of n into exactly 1 part is 1 (just n itself). The number of partitions into exactly 2 parts is floor(n/2). Wait, for example, n=5: floor(5/2)=2, which is 4+1 and 3+2. So total number of partitions into at most 2 parts is 1 + floor(n/2). Therefore, for n=5, it's 1 + 2 = 3. Which matches.

Therefore, the formula is 1 + floor(n/2). Wait, but for n=4, floor(4/2)=2, so 1 + 2=3, which is correct. For n=3, floor(3/2)=1, so 1 +1=2. Correct. For n=2, floor(2/2)=1, so 1 +1=2. Correct. For n=1, floor(1/2)=0, so 1 +0=1. Correct.

Therefore, in general, the number of ways to put n indistinct balls into 2 indistinct boxes is 1 + floor(n/2). Wait, but hold on, for n=5, floor(5/2)=2, so 1 + 2=3. Which is correct.

But let me check for n=6. Then floor(6/2)=3, so 1 + 3=4. Let's list them: 6+0, 5+1, 4+2, 3+3. That's 4, which is correct.

Therefore, the formula seems to hold. Therefore, for n=5, the answer is 3.

But let me think again if there's another way to approach this problem.

Another approach: Since the boxes are indistinct, the problem is equivalent to finding the number of distinct multisets of size 2 where the elements are non-negative integers that sum to 5.

But since the boxes are indistinct, the order doesn't matter. So the number of such multisets is equal to the number of integer solutions (a, b) with a ≥ b ≥ 0 and a + b = 5.

So solving for a and b:

a + b = 5, with a ≥ b ≥ 0.

Let me find all such pairs:

Start with b=0: then a=5. So (5,0)

b=1: a=4. (4,1)

b=2: a=3. (3,2)

b=3: a=2. But since a ≥ b, then a=2 and b=3 would not satisfy a ≥ b. So we stop here.

Therefore, the pairs are (5,0), (4,1), (3,2). So three solutions. Therefore, the answer is 3.

Therefore, regardless of the method, I get 3.

But let me check if there's a possibility that I'm missing something. For example, if the boxes are allowed to be empty? Wait, the problem says "put 5 balls in 2 boxes". It doesn't specify whether boxes can be empty or not. But in standard combinatorial problems, unless specified otherwise, boxes can be empty.

But in our case, since we have considered distributions like (5,0), which includes empty boxes, so that's okay.

Alternatively, if the problem had said that each box must contain at least one ball, then the answer would be different. For example, if we had to put 5 balls into 2 boxes with each box having at least one ball, then the number of distributions would be the number of partitions of 5 into exactly 2 parts, which is floor(5/2)=2. But since the problem doesn't specify that boxes must be non-empty, we include the case where one box is empty.

Therefore, the answer is 3.

But let me think again if there's a different interpretation. For example, sometimes in problems with indistinct boxes, people might consider that putting all balls in one box is not allowed, but I don't think that's the case here. The problem says "put 5 balls in 2 boxes", so it's allowed to have some boxes empty.

Alternatively, maybe the problem is considering that boxes are distinguishable? Wait, no, the problem says neither are the boxes. So they are indistinct.

Alternatively, maybe I need to use Stirling numbers of the second kind? Wait, the Stirling numbers of the second kind count the number of ways to partition a set of n distinguishable objects into k non-empty indistinct subsets. But in our case, the balls are indistinct. So Stirling numbers might not apply here.

Wait, if the balls were distinguishable and boxes are indistinct, then the number of ways would be the Stirling numbers of the second kind S(n, k). But since the balls are indistinct, it's different.

So, for example, if the balls are indistinct and boxes are indistinct, the number of ways is equal to the number of integer partitions of n into at most k parts. Which is what we discussed earlier.

Therefore, for n=5, k=2, it's 3.

Alternatively, if the balls are distinguishable and boxes are indistinct, then it's S(5,2) + S(5,1). Wait, no. Wait, the number of ways to put n distinguishable objects into k indistinct boxes is equal to the sum_{i=1}^k S(n,i). But if we allow empty boxes, then it's the same as the number of partitions of the set into at most k non-empty subsets.

But in our problem, the balls are indistinct, so it's not about set partitions.

Therefore, going back, I think my original reasoning is correct.

Another way to think about it: since the boxes are indistinct, the problem is equivalent to finding the number of distinct ways to split the balls into two groups, where the order of the groups doesn't matter.

So, for example, splitting into group sizes 0 and 5, 1 and 4, 2 and 3. Since 0 and 5 is the same as 5 and 0, and 1 and 4 is same as 4 and 1, etc. So there are three distinct ways.

Therefore, the answer is 3.

But just to make sure, let me think of a concrete example. Suppose I have 5 balls: let's say they are all the same. Let me imagine putting them into two boxes.

Case 1: All 5 in one box, the other is empty. Since boxes are the same, there's only one way.

Case 2: 4 in one box, 1 in the other. Again, since boxes are the same, there's only one way.

Case 3: 3 in one box, 2 in the other. Again, only one way.

No other possibilities. For example, if I tried to do 2 and 3, that's same as 3 and 2. 1 and 4 same as 4 and 1. 0 and 5 same as 5 and 0. So there are only three distinct ways.

Therefore, the answer is 3.

Alternatively, if I think of generating all possible distributions for distinguishable boxes and then grouping them into equivalence classes under swapping boxes.

For distinguishable boxes, the distributions are:

(0,5), (1,4), (2,3), (3,2), (4,1), (5,0).

Now, grouping them into equivalence classes where (a,b) is equivalent to (b,a).

So:

- (0,5) and (5,0) form one class.

- (1,4) and (4,1) form another class.

- (2,3) and (3,2) form another class.

Therefore, there are 3 equivalence classes, hence 3 ways.

Therefore, the answer is 3.

Therefore, I think I can be confident that the answer is 3.

**Final Answer**
The number of ways is \boxed{3}.

To determine how many ways there are to put **5 indistinguishable balls** into **2 indistinguishable boxes**, we need to consider the nature of both the balls and the boxes.

---

### Key Observations:

- **Balls are indistinguishable**: This means that the identity of the balls does not matter. Only the number of balls in each box matters.
- **Boxes are indistinguishable**: This means that swapping the boxes does not result in a new configuration. For example, putting 1 ball in Box A and 4 in Box B is the same as putting 1 in Box B and 4 in Box A.

---

### Reformulating the Problem:

We are essentially looking for the number of **distinct partitions** of the number 5 into **at most 2 parts**, where the order of the parts does not matter. This is because each part represents the number of balls in a box, and the two parts represent the two boxes.

---

### Listing the Partitions:

We list all the integer partitions of 5 into **at most 2 parts**:

1. **5 + 0**
2. **4 + 1**
3. **3 + 2**

These are the only distinct ways to split 5 into two non-negative integers, where the order does not matter. Each of these corresponds to a unique way of distributing the balls into the boxes.

---

### Final Count:

There are **3** such partitions, and since the boxes are indistinguishable, each of these corresponds to a **unique configuration**.

---

### Final Answer:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways there are to put 5 indistinguishable balls into 2 indistinguishable boxes. Hmm, let me think.

First, I remember that when dealing with problems involving distributing objects into boxes, the key factors are whether the objects are distinguishable and whether the boxes are distinguishable. In this case, both the balls and the boxes are not distinguishable. That probably means that the order of the boxes doesn't matter, and the balls are all the same.

Let me recall some combinatorial concepts. If the boxes were distinguishable, like Box A and Box B, then the problem would be simpler. For each ball, you could choose which box it goes into. Since the balls are indistinct, the number of ways would be the number of non-negative integer solutions to the equation:

b1 + b2 = 5

where b1 is the number of balls in box 1 and b2 is the number in box 2. The formula for that is C(n + k - 1, k - 1), where n is the number of balls and k is the number of boxes. So here, that would be C(5 + 2 - 1, 2 - 1) = C(6,1) = 6. But wait, that's if the boxes are distinguishable. However, in our problem, the boxes are not distinguishable. So some of these distributions might be counted multiple times in the distinguishable case, and we need to adjust for that.

Let me think. If the boxes are indistinct, then putting 1 ball in box A and 4 in box B is the same as putting 1 in box B and 4 in box A. So we need to count the number of distinct distributions where the order doesn't matter.

So, for the case of indistinct boxes, the number of ways is equal to the number of integer partitions of 5 into at most 2 parts. Since we have two boxes, each partition will have two parts (which can be equal or not).

Let me list all the possible partitions of 5 into at most 2 parts:

1. 5 + 0
2. 4 + 1
3. 3 + 2

Wait, that's three partitions. But hold on, if we consider that the boxes are indistinct, then partitions like 0 + 5 are the same as 5 + 0, so we don't count them separately. Similarly, 1 + 4 is same as 4 + 1, and 2 + 3 is same as 3 + 2. So the distinct distributions are just those three.

But wait, is that all? Let me check. For example, if we have two boxes, the number of ways to distribute 5 indistinct balls is equal to the floor of (n + 1)/2? Wait, no. Wait, for n=5, the number of partitions into at most 2 parts is equal to the number of integer partitions of 5 where the number of parts is 1 or 2.

But since the boxes are indistinct, we can have:

- All balls in one box: that's 1 way (since both boxes are the same, putting all in one is the same regardless of which box you choose).

- Splitting into two boxes: the number of ways is equal to the number of ways to split 5 into two positive integers where order doesn't matter. Which would be floor(5/2) = 2? Wait, let me think again.

For splitting into two positive integers, the number of partitions is equal to the floor of n/2. Wait, for n=5, the splits are 1+4 and 2+3. So that's two splits. So total number of ways would be 1 (all in one box) + 2 (split into two boxes) = 3.

So that would be 3 ways. Therefore, the answer is 3? But let me verify this with another approach to make sure.

Alternatively, since the boxes are indistinct, the problem reduces to finding the number of equivalence classes under the action of swapping boxes. In the case of distinguishable boxes, we have 6 distributions. But when boxes are indistinct, each distribution is counted twice except when the distribution is symmetric (i.e., when both boxes have the same number of balls).

So, in the case of 5 balls and 2 boxes, how many symmetric distributions are there? Well, for the number of balls in each box to be equal, we need 5 to be even, but 5 is odd. Therefore, there are no symmetric distributions. Therefore, all distributions are counted twice in the distinguishable case. Therefore, the number of distinct distributions is 6 / 2 = 3.

Yes, that matches the previous result. Therefore, the answer is 3.

But let me think again. Let me list all possibilities.

If the boxes are indistinct, then the possible distributions are:

1. All 5 balls in one box, and 0 in the other. Since boxes are indistinct, there's only one way to do this.

2. 4 balls in one box, 1 in the other. Again, since boxes are indistinct, this is only one way.

3. 3 balls in one box, 2 in the other. Again, only one way.

So that's three different ways.

Alternatively, if I think of the problem as the number of ways to partition the integer 5 into up to 2 parts, which is exactly what I did before.

But let me check with another method. Suppose I use the formula for the number of ways to distribute n indistinct objects into k indistinct boxes, which is equal to the number of integer partitions of n into at most k parts.

In this case, n=5, k=2. The number of integer partitions of 5 into at most 2 parts is indeed 3, as listed above.

Alternatively, if I recall that the number of ways is equal to the floor(n/2) + 1? Wait, for n=5, floor(5/2) is 2, plus 1 is 3. That works. For example, if n=4, floor(4/2) +1 = 2 +1=3. Let's check: partitions of 4 into at most 2 parts: 4+0, 3+1, 2+2. That's 3, which matches. So yes, the formula seems to be floor(n/2) +1? Wait, but for n=3, floor(3/2) +1=1+1=2. Partitions of 3 into at most 2 parts: 3+0, 2+1. That's 2, which matches. For n=2: floor(2/2)+1=1+1=2. Partitions: 2+0, 1+1. Correct. For n=1: floor(1/2)+1=0+1=1. Partition: 1+0. Correct. So yes, the formula seems to be floor(n/2) +1? Wait, but for n=5, floor(5/2)=2, so 2+1=3. So that works.

But wait, actually, the formula for the number of partitions of n into at most k parts is not exactly floor(n/2) +1. Wait, that formula seems to be specific for k=2. For general k, it's more complicated. But in our case, since k=2, the number of partitions of n into at most 2 parts is equal to the number of integer partitions of n where the largest part is at most 2. Wait, no, actually, the number of partitions of n into at most 2 parts is equal to the number of partitions of n where the number of parts is at most 2. Which is equivalent to the number of partitions of n into 1 or 2 parts.

But for the number of partitions of n into exactly 1 part is 1 (just n itself). The number of partitions into exactly 2 parts is floor(n/2). Wait, for example, n=5: floor(5/2)=2, which is 4+1 and 3+2. So total number of partitions into at most 2 parts is 1 + floor(n/2). Therefore, for n=5, it's 1 + 2 = 3. Which matches.

Therefore, the formula is 1 + floor(n/2). Wait, but for n=4, floor(4/2)=2, so 1 + 2=3, which is correct. For n=3, floor(3/2)=1, so 1 +1=2. Correct. For n=2, floor(2/2)=1, so 1 +1=2. Correct. For n=1, floor(1/2)=0, so 1 +0=1. Correct.

Therefore, in general, the number of ways to put n indistinct balls into 2 indistinct boxes is 1 + floor(n/2). Wait, but hold on, for n=5, floor(5/2)=2, so 1 + 2=3. Which is correct.

But let me check for n=6. Then floor(6/2)=3, so 1 + 3=4. Let's list them: 6+0, 5+1, 4+2, 3+3. That's 4, which is correct.

Therefore, the formula seems to hold. Therefore, for n=5, the answer is 3.

But let me think again if there's another way to approach this problem.

Another approach: Since the boxes are indistinct, the problem is equivalent to finding the number of distinct multisets of size 2 where the elements are non-negative integers that sum to 5.

But since the boxes are indistinct, the order doesn't matter. So the number of such multisets is equal to the number of integer solutions (a, b) with a ≥ b ≥ 0 and a + b = 5.

So solving for a and b:

a + b = 5, with a ≥ b ≥ 0.

Let me find all such pairs:

Start with b=0: then a=5. So (5,0)

b=1: a=4. (4,1)

b=2: a=3. (3,2)

b=3: a=2. But since a ≥ b, then a=2 and b=3 would not satisfy a ≥ b. So we stop here.

Therefore, the pairs are (5,0), (4,1), (3,2). So three solutions. Therefore, the answer is 3.

Therefore, regardless of the method, I get 3.

But let me check if there's a possibility that I'm missing something. For example, if the boxes are allowed to be empty? Wait, the problem says "put 5 balls in 2 boxes". It doesn't specify whether boxes can be empty or not. But in standard combinatorial problems, unless specified otherwise, boxes can be empty.

But in our case, since we have considered distributions like (5,0), which includes empty boxes, so that's okay.

Alternatively, if the problem had said that each box must contain at least one ball, then the answer would be different. For example, if we had to put 5 balls into 2 boxes with each box having at least one ball, then the number of distributions would be the number of partitions of 5 into exactly 2 parts, which is floor(5/2)=2. But since the problem doesn't specify that boxes must be non-empty, we include the case where one box is empty.

Therefore, the answer is 3.

But let me think again if there's a different interpretation. For example, sometimes in problems with indistinct boxes, people might consider that putting all balls in one box is not allowed, but I don't think that's the case here. The problem says "put 5 balls in 2 boxes", so it's allowed to have some boxes empty.

Alternatively, maybe the problem is considering that boxes are distinguishable? Wait, no, the problem says neither are the boxes. So they are indistinct.

Alternatively, maybe I need to use Stirling numbers of the second kind? Wait, the Stirling numbers of the second kind count the number of ways to partition a set of n distinguishable objects into k non-empty indistinct subsets. But in our case, the balls are indistinct. So Stirling numbers might not apply here.

Wait, if the balls were distinguishable and boxes are indistinct, then the number of ways would be the Stirling numbers of the second kind S(n, k). But since the balls are indistinct, it's different.

So, for example, if the balls are indistinct and boxes are indistinct, the number of ways is equal to the number of integer partitions of n into at most k parts. Which is what we discussed earlier.

Therefore, for n=5, k=2, it's 3.

Alternatively, if the balls are distinguishable and boxes are indistinct, then it's S(5,2) + S(5,1). Wait, no. Wait, the number of ways to put n distinguishable objects into k indistinct boxes is equal to the sum_{i=1}^k S(n,i). But if we allow empty boxes, then it's the same as the number of partitions of the set into at most k non-empty subsets.

But in our problem, the balls are indistinct, so it's not about set partitions.

Therefore, going back, I think my original reasoning is correct.

Another way to think about it: since the boxes are indistinct, the problem is equivalent to finding the number of distinct ways to split the balls into two groups, where the order of the groups doesn't matter.

So, for example, splitting into group sizes 0 and 5, 1 and 4, 2 and 3. Since 0 and 5 is the same as 5 and 0, and 1 and 4 is same as 4 and 1, etc. So there are three distinct ways.

Therefore, the answer is 3.

But just to make sure, let me think of a concrete example. Suppose I have 5 balls: let's say they are all the same. Let me imagine putting them into two boxes.

Case 1: All 5 in one box, the other is empty. Since boxes are the same, there's only one way.

Case 2: 4 in one box, 1 in the other. Again, since boxes are the same, there's only one way.

Case 3: 3 in one box, 2 in the other. Again, only one way.

No other possibilities. For example, if I tried to do 2 and 3, that's same as 3 and 2. 1 and 4 same as 4 and 1. 0 and 5 same as 5 and 0. So there are only three distinct ways.

Therefore, the answer is 3.

Alternatively, if I think of generating all possible distributions for distinguishable boxes and then grouping them into equivalence classes under swapping boxes.

For distinguishable boxes, the distributions are:

(0,5), (1,4), (2,3), (3,2), (4,1), (5,0).

Now, grouping them into equivalence classes where (a,b) is equivalent to (b,a).

So:

- (0,5) and (5,0) form one class.

- (1,4) and (4,1) form another class.

- (2,3) and (3,2) form another class.

Therefore, there are 3 equivalence classes, hence 3 ways.

Therefore, the answer is 3.

Therefore, I think I can be confident that the answer is 3.

**Final Answer**
The number of ways is \boxed{3}.

To determine how many ways there are to put **5 indistinguishable balls** into **2 indistinguishable boxes**, we need to consider the nature of both the balls and the boxes.

---

### Key Observations:

- **Balls are indistinguishable**: This means that the identity of the balls does not matter. Only the number of balls in each box matters.
- **Boxes are indistinguishable**: This means that swapping the boxes does not result in a new configuration. For example, putting 1 ball in Box A and 4 in Box B is the same as putting 1 in Box B and 4 in Box A.

---

### Reformulating the Problem:

We are essentially looking for the number of **distinct partitions** of the number 5 into **at most 2 parts**, where the order of the parts does not matter. This is because each part represents the number of balls in a box, and the two parts represent the two boxes.

---

### Listing the Partitions:

We list all the integer partitions of 5 into **at most 2 parts**:

1. **5 + 0**
2. **4 + 1**
3. **3 + 2**

These are the only distinct ways to split 5 into two non-negative integers, where the order does not matter. Each of these corresponds to a unique way of distributing the balls into the boxes.

---

### Final Count:

There are **3** such partitions, and since the boxes are indistinguishable, each of these corresponds to a **unique configuration**.

---

### Final Answer:

$$
\boxed{3}
$$

Solution: Since both the balls and boxes are indistinguishable, we can arrange them with 5 in one and 0 in the other, 4 in one and 1 in the other, or 3 in one and 2 in the other, for a total of $\boxed{3}$ different arrangements.


Explanation:
Okay, so I need to figure out what shape the equation z = c represents in cylindrical coordinates. Let me start by recalling what cylindrical coordinates are. In cylindrical coordinates, a point in 3D space is represented by (r, θ, z), where r is the radial distance from the z-axis, θ is the angle around the z-axis, and z is the height, just like in Cartesian coordinates.

The equation given is z = c, where c is a constant. So, in cylindrical coordinates, this equation is saying that the z-coordinate is fixed at c, regardless of the values of r and θ. Let me think about what that means geometrically.

In Cartesian coordinates, the equation z = c would represent a plane that's parallel to the xy-plane and intersects the z-axis at z = c. But since we're dealing with cylindrical coordinates here, I need to make sure that this equation translates to the same shape or something different.

Wait, cylindrical coordinates are just a different way of representing points in 3D space. So, if z is fixed, then regardless of how r and θ vary, the z-coordinate remains the same. So, for every point on this surface, the height above the xy-plane is constant. That should still be a plane, right? Because in Cartesian coordinates, z = c is a plane, and since cylindrical coordinates are just a coordinate system that can represent the same space, the equation z = c should still represent a plane.

But let me double-check. Let me think about how cylindrical coordinates relate to Cartesian coordinates. The conversion formulas are:

x = r cos θ

y = r sin θ

z = z

So, if we have z = c, then in Cartesian coordinates, that's exactly z = c. Therefore, the equation z = c in cylindrical coordinates is equivalent to z = c in Cartesian coordinates, which is a plane.

But wait, maybe there's a trick here? Sometimes equations in cylindrical coordinates can represent different shapes. For example, if you have r = c, that's a cylinder, right? Because r is the distance from the z-axis, so all points at a fixed distance from the z-axis form a cylinder. Similarly, if you have θ = c, that's a plane that goes through the z-axis, like a vertical plane.

But in this case, z = c is not involving r or θ. So, it's just fixing the z-coordinate. So, in cylindrical coordinates, z is the same as in Cartesian coordinates. Therefore, z = c is a plane.

But let me think again. Suppose I fix z = c. Then, in cylindrical coordinates, r and θ can be anything. So, for example, if I fix z = 2, then any point with z = 2, regardless of how far away from the z-axis (so any r) and any angle θ, is part of this surface. So, that's like a flat plane extending infinitely in the radial and angular directions.

So, in other words, it's a horizontal plane cutting through the z-axis at height c. So, that's a plane. Therefore, the answer should be (C) Plane.

But let me check if there's any possibility that it could be something else. For example, if the equation was r = c, that's a cylinder. If it was θ = c, that's a plane. But here, z is fixed. So, in cylindrical coordinates, z is the same as in Cartesian coordinates. So, z = c is a plane.

Alternatively, maybe someone might confuse it with a horizontal cylinder? But no, a cylinder would require r to be fixed. For example, if you have r = c, then that's a cylinder with radius c. But here, z is fixed.

Alternatively, if you have z = r, that would be a cone. But here, z is fixed.

So, I think the answer is (C) Plane.

But let me check if there's any other way to think about it. For example, in cylindrical coordinates, if you have z = c, then in terms of Cartesian coordinates, that's z = c. So, the shape is a plane. Therefore, the answer is (C).

Alternatively, if someone thought that in cylindrical coordinates, z = c might be a line or something else, but no. Because in cylindrical coordinates, z is independent of r and θ. So, for each z = c, you can have any r and θ. So, that's an infinite plane.

Therefore, the answer is (C) Plane.

**Final Answer**
\boxed{C}

To determine the shape described by the equation
$$
z = c
$$
in **cylindrical coordinates** $(r, \theta, z)$, we begin by understanding the nature of cylindrical coordinates and how they relate to the given equation.

---

### Understanding Cylindrical Coordinates

In cylindrical coordinates, a point in 3D space is represented as $(r, \theta, z)$, where:

- $r$ is the radial distance from the $z$-axis,
- $\theta$ is the angle measured counterclockwise from the positive $x$-axis in the $xy$-plane,
- $z$ is the height (or vertical coordinate), identical to the Cartesian $z$-coordinate.

This means that the coordinate $z$ in cylindrical coordinates is **exactly the same** as the $z$-coordinate in Cartesian coordinates.

---

### Interpreting the Equation $z = c$

The equation $z = c$ fixes the $z$-coordinate to a constant value $c$, regardless of the values of $r$ and $\theta$. This implies that:

- The height of every point on the surface is the same: $z = c$,
- The radial distance $r$ and angular position $\theta$ can vary freely.

This is equivalent to saying that the surface includes all points at height $c$, no matter how far they are from the $z$-axis or in which direction they are oriented around it.

---

### Geometric Interpretation

In Cartesian coordinates, the equation $z = c$ represents a **horizontal plane** that is parallel to the $xy$-plane and intersects the $z$-axis at height $z = c$. Since cylindrical coordinates are just a different representation of the same 3D space, the equation $z = c$ in cylindrical coordinates describes the **same** geometric object: a **plane**.

This is in contrast to other equations in cylindrical coordinates:

- $r = c$ describes a **cylinder** (a surface of constant radius),
- $\theta = c$ describes a **vertical plane** through the $z$-axis,
- $z = r$ describes a **cone**.

But in this case, since only $z$ is fixed and $r$ and $\theta$ are free to vary, the shape is clearly a **plane**.

---

### Final Answer

$$
\boxed{C}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what shape the equation z = c represents in cylindrical coordinates. Let me start by recalling what cylindrical coordinates are. In cylindrical coordinates, a point in 3D space is represented by (r, θ, z), where r is the radial distance from the z-axis, θ is the angle around the z-axis, and z is the height, just like in Cartesian coordinates.

The equation given is z = c, where c is a constant. So, in cylindrical coordinates, this equation is saying that the z-coordinate is fixed at c, regardless of the values of r and θ. Let me think about what that means geometrically.

In Cartesian coordinates, the equation z = c would represent a plane that's parallel to the xy-plane and intersects the z-axis at z = c. But since we're dealing with cylindrical coordinates here, I need to make sure that this equation translates to the same shape or something different.

Wait, cylindrical coordinates are just a different way of representing points in 3D space. So, if z is fixed, then regardless of how r and θ vary, the z-coordinate remains the same. So, for every point on this surface, the height above the xy-plane is constant. That should still be a plane, right? Because in Cartesian coordinates, z = c is a plane, and since cylindrical coordinates are just a coordinate system that can represent the same space, the equation z = c should still represent a plane.

But let me double-check. Let me think about how cylindrical coordinates relate to Cartesian coordinates. The conversion formulas are:

x = r cos θ

y = r sin θ

z = z

So, if we have z = c, then in Cartesian coordinates, that's exactly z = c. Therefore, the equation z = c in cylindrical coordinates is equivalent to z = c in Cartesian coordinates, which is a plane.

But wait, maybe there's a trick here? Sometimes equations in cylindrical coordinates can represent different shapes. For example, if you have r = c, that's a cylinder, right? Because r is the distance from the z-axis, so all points at a fixed distance from the z-axis form a cylinder. Similarly, if you have θ = c, that's a plane that goes through the z-axis, like a vertical plane.

But in this case, z = c is not involving r or θ. So, it's just fixing the z-coordinate. So, in cylindrical coordinates, z is the same as in Cartesian coordinates. Therefore, z = c is a plane.

But let me think again. Suppose I fix z = c. Then, in cylindrical coordinates, r and θ can be anything. So, for example, if I fix z = 2, then any point with z = 2, regardless of how far away from the z-axis (so any r) and any angle θ, is part of this surface. So, that's like a flat plane extending infinitely in the radial and angular directions.

So, in other words, it's a horizontal plane cutting through the z-axis at height c. So, that's a plane. Therefore, the answer should be (C) Plane.

But let me check if there's any possibility that it could be something else. For example, if the equation was r = c, that's a cylinder. If it was θ = c, that's a plane. But here, z is fixed. So, in cylindrical coordinates, z is the same as in Cartesian coordinates. So, z = c is a plane.

Alternatively, maybe someone might confuse it with a horizontal cylinder? But no, a cylinder would require r to be fixed. For example, if you have r = c, then that's a cylinder with radius c. But here, z is fixed.

Alternatively, if you have z = r, that would be a cone. But here, z is fixed.

So, I think the answer is (C) Plane.

But let me check if there's any other way to think about it. For example, in cylindrical coordinates, if you have z = c, then in terms of Cartesian coordinates, that's z = c. So, the shape is a plane. Therefore, the answer is (C).

Alternatively, if someone thought that in cylindrical coordinates, z = c might be a line or something else, but no. Because in cylindrical coordinates, z is independent of r and θ. So, for each z = c, you can have any r and θ. So, that's an infinite plane.

Therefore, the answer is (C) Plane.

**Final Answer**
\boxed{C}

To determine the shape described by the equation
$$
z = c
$$
in **cylindrical coordinates** $(r, \theta, z)$, we begin by understanding the nature of cylindrical coordinates and how they relate to the given equation.

---

### Understanding Cylindrical Coordinates

In cylindrical coordinates, a point in 3D space is represented as $(r, \theta, z)$, where:

- $r$ is the radial distance from the $z$-axis,
- $\theta$ is the angle measured counterclockwise from the positive $x$-axis in the $xy$-plane,
- $z$ is the height (or vertical coordinate), identical to the Cartesian $z$-coordinate.

This means that the coordinate $z$ in cylindrical coordinates is **exactly the same** as the $z$-coordinate in Cartesian coordinates.

---

### Interpreting the Equation $z = c$

The equation $z = c$ fixes the $z$-coordinate to a constant value $c$, regardless of the values of $r$ and $\theta$. This implies that:

- The height of every point on the surface is the same: $z = c$,
- The radial distance $r$ and angular position $\theta$ can vary freely.

This is equivalent to saying that the surface includes all points at height $c$, no matter how far they are from the $z$-axis or in which direction they are oriented around it.

---

### Geometric Interpretation

In Cartesian coordinates, the equation $z = c$ represents a **horizontal plane** that is parallel to the $xy$-plane and intersects the $z$-axis at height $z = c$. Since cylindrical coordinates are just a different representation of the same 3D space, the equation $z = c$ in cylindrical coordinates describes the **same** geometric object: a **plane**.

This is in contrast to other equations in cylindrical coordinates:

- $r = c$ describes a **cylinder** (a surface of constant radius),
- $\theta = c$ describes a **vertical plane** through the $z$-axis,
- $z = r$ describes a **cone**.

But in this case, since only $z$ is fixed and $r$ and $\theta$ are free to vary, the shape is clearly a **plane**.

---

### Final Answer

$$
\boxed{C}
$$

Solution:
In cylindrical coordinates, $z$ simply denotes the $z$-coordinate of a point.  Thus, for a fixed $z$-coordinate $c,$ all the points lie on a plane that is parallel to the $xy$-plane.  The answer is $\boxed{\text{(C)}}.$

[asy]
import three;
import solids;

size(200);
currentprojection = perspective(6,3,2);
currentlight = (1,0,1);
real theta = 120;

draw((-2,0,0)--(2,0,0));
draw((0,-2,0)--(0,2,0));
draw(surface((1,1,0.5)--(1,-1,0.5)--(-1,-1,0.5)--(-1,1,0.5)--cycle),gray(0.99));
draw((0,0,-2)--(0,0,0.2));
draw((0,0,0.5)--(0,0,2));

label("$x$", (2,0,0), SW);
label("$y$", (0,2,0), E);
label("$z$", (0,0,2), N);
label("$z = c$", (-1,1,0.5), E);
[/asy]


Explanation:
Okay, so I need to find the sum of the roots of the function g(x), which is defined as f(x + 5). The original function f(x) is given as x³ - 49x² + 623x - 2015. Hmm, let me recall how to approach this.

First, I remember that for a polynomial equation, the sum of the roots can be found using Vieta's formulas. For a cubic equation of the form ax³ + bx² + cx + d = 0, the sum of the roots is -b/a. So, if I can figure out the coefficients of g(x), then I can apply Vieta's formula directly.

But wait, g(x) is f(x + 5). That means I need to find the polynomial expression for g(x) by substituting (x + 5) into f. However, expanding that might be tedious, but maybe there's a smarter way. Let me think.

Alternatively, since g(x) = f(x + 5), the roots of g(x) are the values of x such that f(x + 5) = 0. Let me denote y = x + 5. Then, the equation becomes f(y) = 0. Therefore, the roots of g(x) are the roots of f(y) = 0 shifted by -5. In other words, if the roots of f(y) = 0 are r₁, r₂, r₃, then the roots of g(x) = 0 are r₁ - 5, r₂ - 5, r₃ - 5.

Therefore, the sum of the roots of g(x) would be (r₁ - 5) + (r₂ - 5) + (r₃ - 5) = (r₁ + r₂ + r₃) - 15.

But from Vieta's formula on the original polynomial f(y) = y³ - 49y² + 623y - 2015, the sum of the roots r₁ + r₂ + r₃ is equal to -(-49)/1 = 49. Therefore, the sum of the roots of g(x) would be 49 - 15 = 34.

Wait, is that correct? Let me check again.

Yes, because if the original polynomial f(y) has roots r₁, r₂, r₃, then the sum is 49. Then shifting each root by -5, so subtracting 5 from each root, so total sum is 49 - 15 = 34. Therefore, the answer is 34?

But let me confirm this by another method, just to be sure. Maybe expanding g(x) and then applying Vieta's formula directly. Let's try that.

Given that g(x) = f(x + 5) = (x + 5)³ - 49(x + 5)² + 623(x + 5) - 2015. Let me expand this step by step.

First, expand (x + 5)³:

(x + 5)³ = x³ + 3*5x² + 3*5²x + 5³ = x³ + 15x² + 75x + 125.

Next, expand -49(x + 5)²:

(x + 5)² = x² + 10x + 25, so multiplying by -49 gives -49x² - 490x - 1225.

Then, expand 623(x + 5):

That's 623x + 3115.

Now, putting all together:

g(x) = [x³ + 15x² + 75x + 125] + [-49x² - 490x - 1225] + [623x + 3115] - 2015.

Now combine like terms:

Start with the x³ term: 1x³.

For x² terms: 15x² - 49x² = -34x².

For x terms: 75x - 490x + 623x. Let me compute that:

75 - 490 = -415; -415 + 623 = 208. So 208x.

Constant terms: 125 - 1225 + 3115 - 2015.

Let me compute step by step:

125 - 1225 = -1100.

-1100 + 3115 = 2015.

2015 - 2015 = 0.

So the constant term is 0.

Therefore, g(x) simplifies to x³ - 34x² + 208x.

Wait, so g(x) = x³ - 34x² + 208x. Therefore, the polynomial is x³ - 34x² + 208x.

But then, if we write it in standard form, it's x³ - 34x² + 208x + 0. Therefore, the coefficients are a = 1, b = -34, c = 208, d = 0.

Therefore, according to Vieta's formula, the sum of the roots is -b/a = -(-34)/1 = 34. Which matches the previous result.

So that's reassuring. Therefore, the sum of the roots of g(x) is 34.

But let me check once more. Let me think if there's another way to see this.

Alternatively, since shifting the function horizontally affects the roots. If you have a function f(x) with roots r₁, r₂, r₃, then g(x) = f(x + 5) has roots at x = r₁ - 5, x = r₂ - 5, x = r₃ - 5. Therefore, the sum of the roots is (r₁ + r₂ + r₃) - 15.

As before, the original sum of roots of f(x) is 49, so 49 - 15 = 34.

Therefore, both methods give the same answer. So I think that's solid.

But just to be thorough, let me check if expanding was done correctly. Let me recheck the expansion step by step.

Original expansion:

First term: (x + 5)^3 = x³ + 15x² + 75x + 125. Correct.

Second term: -49(x + 5)^2. Let me compute (x + 5)^2 again: x² + 10x + 25. Multiply by -49: -49x² - 490x - 1225. Correct.

Third term: 623(x + 5) = 623x + 3115. Correct.

Fourth term: -2015.

Now adding all together:

Start with x³ + 15x² + 75x + 125 - 49x² - 490x - 1225 + 623x + 3115 - 2015.

Combine x³: 1x³.

x² terms: 15x² - 49x² = -34x². Correct.

x terms: 75x - 490x + 623x. Let me compute 75 - 490 = -415; -415 + 623 = 208. So 208x. Correct.

Constants: 125 - 1225 + 3115 - 2015. Let me compute:

Start with 125 - 1225: that's -1100.

Then -1100 + 3115: 3115 - 1100 = 2015.

Then 2015 - 2015 = 0. Correct.

So the polynomial is indeed x³ - 34x² + 208x. Therefore, the sum of roots is 34.

Alternatively, maybe I can factor g(x). Let me see.

g(x) = x³ - 34x² + 208x = x(x² - 34x + 208).

So the roots are x = 0, and roots of x² - 34x + 208 = 0.

But if I compute the sum of roots, it's 0 + [34/1] (from quadratic) = 34. Which again gives the same result.

Therefore, regardless of the method, the answer is 34.

But just to make sure, let me check if the original polynomial f(x) has roots that when shifted by -5 would sum to 34.

But since we already know that the original sum of roots is 49, and shifting each root by -5 subtracts 5 from each, so total subtracts 15, so 49 - 15 = 34.

Alternatively, if I wanted to find the roots of f(x) and then shift them, but since f(x) is a cubic, it's hard to find exact roots, but maybe I can check with Vieta's formula.

But since Vieta's formula is straightforward, and we have already confirmed with two methods, I think that's solid.

Therefore, I can be confident that the sum of the roots of g(x) is 34.

**Final Answer**
\boxed{34}

To find the sum of the roots of the function $ g(x) = f(x + 5) $, where $ f(x) = x^3 - 49x^2 + 623x - 2015 $, we can use **Vieta's formulas** and properties of polynomial transformations.

---

### Step 1: Understand the Relationship Between $ f(x) $ and $ g(x) $

The function $ g(x) = f(x + 5) $ is a horizontal shift of the function $ f(x) $ to the left by 5 units. This means that if $ r_1, r_2, r_3 $ are the roots of $ f(x) = 0 $, then the roots of $ g(x) = 0 $ are:

$$
r_1 - 5, \quad r_2 - 5, \quad r_3 - 5
$$

So, the sum of the roots of $ g(x) $ is:

$$
(r_1 - 5) + (r_2 - 5) + (r_3 - 5) = (r_1 + r_2 + r_3) - 15
$$

---

### Step 2: Use Vieta's Formula on $ f(x) $

From Vieta's formulas for a cubic polynomial of the form:

$$
f(x) = x^3 + ax^2 + bx + c
$$

the sum of the roots is:

$$
r_1 + r_2 + r_3 = -\frac{a}{1} = -a
$$

In our case, $ f(x) = x^3 - 49x^2 + 623x - 2015 $, so the coefficient of $ x^2 $ is $ -49 $. Therefore:

$$
r_1 + r_2 + r_3 = -(-49) = 49
$$

---

### Step 3: Compute the Sum of the Roots of $ g(x) $

Now, using the earlier expression:

$$
\text{Sum of roots of } g(x) = (r_1 + r_2 + r_3) - 15 = 49 - 15 = 34
$$

---

### Step 4: Confirm by Expanding $ g(x) $

To double-check, we can expand $ g(x) = f(x + 5) $:

$$
g(x) = (x + 5)^3 - 49(x + 5)^2 + 623(x + 5) - 2015
$$

Expanding each term:

- $ (x + 5)^3 = x^3 + 15x^2 + 75x + 125 $
- $ -49(x + 5)^2 = -49(x^2 + 10x + 25) = -49x^2 - 490x - 1225 $
- $ 623(x + 5) = 623x + 3115 $
- Constant term: $ -2015 $

Now combine all terms:

$$
g(x) = x^3 + 15x^2 + 75x + 125 - 49x^2 - 490x - 1225 + 623x + 3115 - 2015
$$

Group like terms:

- $ x^3 $: $ x^3 $
- $ x^2 $: $ 15x^2 - 49x^2 = -34x^2 $
- $ x $: $ 75x - 490x + 623x = 208x $
- Constants: $ 125 - 1225 + 3115 - 2015 = 0 $

So:

$$
g(x) = x^3 - 34x^2 + 208x
$$

This is a cubic polynomial with leading coefficient 1 and coefficient of $ x^2 $ equal to $ -34 $. Using Vieta's formula again:

$$
\text{Sum of roots} = -\frac{-34}{1} = 34
$$

---

### Final Answer

$$
\boxed{34}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of the roots of the function g(x), which is defined as f(x + 5). The original function f(x) is given as x³ - 49x² + 623x - 2015. Hmm, let me recall how to approach this.

First, I remember that for a polynomial equation, the sum of the roots can be found using Vieta's formulas. For a cubic equation of the form ax³ + bx² + cx + d = 0, the sum of the roots is -b/a. So, if I can figure out the coefficients of g(x), then I can apply Vieta's formula directly.

But wait, g(x) is f(x + 5). That means I need to find the polynomial expression for g(x) by substituting (x + 5) into f. However, expanding that might be tedious, but maybe there's a smarter way. Let me think.

Alternatively, since g(x) = f(x + 5), the roots of g(x) are the values of x such that f(x + 5) = 0. Let me denote y = x + 5. Then, the equation becomes f(y) = 0. Therefore, the roots of g(x) are the roots of f(y) = 0 shifted by -5. In other words, if the roots of f(y) = 0 are r₁, r₂, r₃, then the roots of g(x) = 0 are r₁ - 5, r₂ - 5, r₃ - 5.

Therefore, the sum of the roots of g(x) would be (r₁ - 5) + (r₂ - 5) + (r₃ - 5) = (r₁ + r₂ + r₃) - 15.

But from Vieta's formula on the original polynomial f(y) = y³ - 49y² + 623y - 2015, the sum of the roots r₁ + r₂ + r₃ is equal to -(-49)/1 = 49. Therefore, the sum of the roots of g(x) would be 49 - 15 = 34.

Wait, is that correct? Let me check again.

Yes, because if the original polynomial f(y) has roots r₁, r₂, r₃, then the sum is 49. Then shifting each root by -5, so subtracting 5 from each root, so total sum is 49 - 15 = 34. Therefore, the answer is 34?

But let me confirm this by another method, just to be sure. Maybe expanding g(x) and then applying Vieta's formula directly. Let's try that.

Given that g(x) = f(x + 5) = (x + 5)³ - 49(x + 5)² + 623(x + 5) - 2015. Let me expand this step by step.

First, expand (x + 5)³:

(x + 5)³ = x³ + 3*5x² + 3*5²x + 5³ = x³ + 15x² + 75x + 125.

Next, expand -49(x + 5)²:

(x + 5)² = x² + 10x + 25, so multiplying by -49 gives -49x² - 490x - 1225.

Then, expand 623(x + 5):

That's 623x + 3115.

Now, putting all together:

g(x) = [x³ + 15x² + 75x + 125] + [-49x² - 490x - 1225] + [623x + 3115] - 2015.

Now combine like terms:

Start with the x³ term: 1x³.

For x² terms: 15x² - 49x² = -34x².

For x terms: 75x - 490x + 623x. Let me compute that:

75 - 490 = -415; -415 + 623 = 208. So 208x.

Constant terms: 125 - 1225 + 3115 - 2015.

Let me compute step by step:

125 - 1225 = -1100.

-1100 + 3115 = 2015.

2015 - 2015 = 0.

So the constant term is 0.

Therefore, g(x) simplifies to x³ - 34x² + 208x.

Wait, so g(x) = x³ - 34x² + 208x. Therefore, the polynomial is x³ - 34x² + 208x.

But then, if we write it in standard form, it's x³ - 34x² + 208x + 0. Therefore, the coefficients are a = 1, b = -34, c = 208, d = 0.

Therefore, according to Vieta's formula, the sum of the roots is -b/a = -(-34)/1 = 34. Which matches the previous result.

So that's reassuring. Therefore, the sum of the roots of g(x) is 34.

But let me check once more. Let me think if there's another way to see this.

Alternatively, since shifting the function horizontally affects the roots. If you have a function f(x) with roots r₁, r₂, r₃, then g(x) = f(x + 5) has roots at x = r₁ - 5, x = r₂ - 5, x = r₃ - 5. Therefore, the sum of the roots is (r₁ + r₂ + r₃) - 15.

As before, the original sum of roots of f(x) is 49, so 49 - 15 = 34.

Therefore, both methods give the same answer. So I think that's solid.

But just to be thorough, let me check if expanding was done correctly. Let me recheck the expansion step by step.

Original expansion:

First term: (x + 5)^3 = x³ + 15x² + 75x + 125. Correct.

Second term: -49(x + 5)^2. Let me compute (x + 5)^2 again: x² + 10x + 25. Multiply by -49: -49x² - 490x - 1225. Correct.

Third term: 623(x + 5) = 623x + 3115. Correct.

Fourth term: -2015.

Now adding all together:

Start with x³ + 15x² + 75x + 125 - 49x² - 490x - 1225 + 623x + 3115 - 2015.

Combine x³: 1x³.

x² terms: 15x² - 49x² = -34x². Correct.

x terms: 75x - 490x + 623x. Let me compute 75 - 490 = -415; -415 + 623 = 208. So 208x. Correct.

Constants: 125 - 1225 + 3115 - 2015. Let me compute:

Start with 125 - 1225: that's -1100.

Then -1100 + 3115: 3115 - 1100 = 2015.

Then 2015 - 2015 = 0. Correct.

So the polynomial is indeed x³ - 34x² + 208x. Therefore, the sum of roots is 34.

Alternatively, maybe I can factor g(x). Let me see.

g(x) = x³ - 34x² + 208x = x(x² - 34x + 208).

So the roots are x = 0, and roots of x² - 34x + 208 = 0.

But if I compute the sum of roots, it's 0 + [34/1] (from quadratic) = 34. Which again gives the same result.

Therefore, regardless of the method, the answer is 34.

But just to make sure, let me check if the original polynomial f(x) has roots that when shifted by -5 would sum to 34.

But since we already know that the original sum of roots is 49, and shifting each root by -5 subtracts 5 from each, so total subtracts 15, so 49 - 15 = 34.

Alternatively, if I wanted to find the roots of f(x) and then shift them, but since f(x) is a cubic, it's hard to find exact roots, but maybe I can check with Vieta's formula.

But since Vieta's formula is straightforward, and we have already confirmed with two methods, I think that's solid.

Therefore, I can be confident that the sum of the roots of g(x) is 34.

**Final Answer**
\boxed{34}

To find the sum of the roots of the function $ g(x) = f(x + 5) $, where $ f(x) = x^3 - 49x^2 + 623x - 2015 $, we can use **Vieta's formulas** and properties of polynomial transformations.

---

### Step 1: Understand the Relationship Between $ f(x) $ and $ g(x) $

The function $ g(x) = f(x + 5) $ is a horizontal shift of the function $ f(x) $ to the left by 5 units. This means that if $ r_1, r_2, r_3 $ are the roots of $ f(x) = 0 $, then the roots of $ g(x) = 0 $ are:

$$
r_1 - 5, \quad r_2 - 5, \quad r_3 - 5
$$

So, the sum of the roots of $ g(x) $ is:

$$
(r_1 - 5) + (r_2 - 5) + (r_3 - 5) = (r_1 + r_2 + r_3) - 15
$$

---

### Step 2: Use Vieta's Formula on $ f(x) $

From Vieta's formulas for a cubic polynomial of the form:

$$
f(x) = x^3 + ax^2 + bx + c
$$

the sum of the roots is:

$$
r_1 + r_2 + r_3 = -\frac{a}{1} = -a
$$

In our case, $ f(x) = x^3 - 49x^2 + 623x - 2015 $, so the coefficient of $ x^2 $ is $ -49 $. Therefore:

$$
r_1 + r_2 + r_3 = -(-49) = 49
$$

---

### Step 3: Compute the Sum of the Roots of $ g(x) $

Now, using the earlier expression:

$$
\text{Sum of roots of } g(x) = (r_1 + r_2 + r_3) - 15 = 49 - 15 = 34
$$

---

### Step 4: Confirm by Expanding $ g(x) $

To double-check, we can expand $ g(x) = f(x + 5) $:

$$
g(x) = (x + 5)^3 - 49(x + 5)^2 + 623(x + 5) - 2015
$$

Expanding each term:

- $ (x + 5)^3 = x^3 + 15x^2 + 75x + 125 $
- $ -49(x + 5)^2 = -49(x^2 + 10x + 25) = -49x^2 - 490x - 1225 $
- $ 623(x + 5) = 623x + 3115 $
- Constant term: $ -2015 $

Now combine all terms:

$$
g(x) = x^3 + 15x^2 + 75x + 125 - 49x^2 - 490x - 1225 + 623x + 3115 - 2015
$$

Group like terms:

- $ x^3 $: $ x^3 $
- $ x^2 $: $ 15x^2 - 49x^2 = -34x^2 $
- $ x $: $ 75x - 490x + 623x = 208x $
- Constants: $ 125 - 1225 + 3115 - 2015 = 0 $

So:

$$
g(x) = x^3 - 34x^2 + 208x
$$

This is a cubic polynomial with leading coefficient 1 and coefficient of $ x^2 $ equal to $ -34 $. Using Vieta's formula again:

$$
\text{Sum of roots} = -\frac{-34}{1} = 34
$$

---

### Final Answer

$$
\boxed{34}
$$

Solution:
Let $a,$ $b,$ $c$ be the roots of $x^3 - 49x^2 + 623x - 2015.$  Then by Vieta's formulas, $a + b + c = 49.$

The roots of $g(x) = f(x + 5)$ are $a - 5,$ $b - 5,$ and $c - 5,$ and their sum is $a + b + c - 15 = 49 - 15 = \boxed{34}.$


Explanation:
Okay, so I need to find the value of p(3)/p(4) where p(x) is a quadratic polynomial with integer coefficients and one of its roots is 4 - sqrt(11). Hmm, let me think about how to approach this.

First, since p(x) is a quadratic polynomial with integer coefficients and one of its roots is 4 - sqrt(11), I remember that irrational roots of polynomials with integer coefficients come in conjugate pairs. That means if 4 - sqrt(11) is a root, then its conjugate, which would be 4 + sqrt(11), must also be a root. Is that right? Let me confirm. Yes, because if you have a polynomial with integer coefficients and a root that's a radical like a + sqrt(b), then the conjugate a - sqrt(b) must also be a root to ensure that when you multiply out the factors, the irrational parts cancel out, leaving integer coefficients. So, that makes sense.

Therefore, the quadratic polynomial p(x) can be written as p(x) = (x - (4 - sqrt(11)))(x - (4 + sqrt(11))). Let me check that. If I expand this, it should give me a quadratic with integer coefficients. Let me do the expansion.

First, multiply the two binomials:

(x - 4 + sqrt(11))(x - 4 - sqrt(11)). Wait, actually, the roots are 4 - sqrt(11) and 4 + sqrt(11), so the factors are (x - (4 - sqrt(11))) and (x - (4 + sqrt(11))). Let me write that as (x - 4 + sqrt(11))(x - 4 - sqrt(11)). That looks like a difference of squares. Let me set a = x - 4 and b = sqrt(11). Then the expression becomes (a + b)(a - b) = a² - b². So substituting back, that would be (x - 4)² - (sqrt(11))². Let me compute that.

(x - 4)² is x² - 8x + 16, and (sqrt(11))² is 11. Therefore, the polynomial is x² - 8x + 16 - 11 = x² - 8x + 5. So p(x) = x² - 8x + 5. Wait, but the problem says that p(x) has integer coefficients, which this does. So that seems correct.

But hold on, is this the only possible polynomial? Because any scalar multiple of this polynomial would also have integer coefficients. For example, if we multiply by 2, we get 2x² - 16x + 10, which also has integer coefficients. However, the problem says "a quadratic polynomial with integer coefficients", so there could be multiple polynomials. But since we are asked to compute p(3)/p(4), maybe the ratio is the same regardless of the scalar multiple? Let me check.

Suppose p(x) = k(x² - 8x + 5) where k is an integer. Then p(3) = k(9 - 24 + 5) = k(-10), and p(4) = k(16 - 32 + 5) = k(-11). Therefore, p(3)/p(4) = (-10k)/(-11k) = 10/11. So regardless of the value of k (as long as k ≠ 0, which it can't be because then p(x) would be the zero polynomial, which isn't quadratic), the ratio is 10/11. Therefore, the answer is 10/11. Wait, so even though there are multiple polynomials, the ratio ends up being the same? That seems right because the k cancels out. Therefore, the answer is 10/11.

But let me verify this by actually computing p(3) and p(4) for the specific polynomial I found. Let me take p(x) = x² - 8x + 5. Then p(3) is 3² - 8*3 + 5 = 9 - 24 + 5 = (9 + 5) - 24 = 14 - 24 = -10. Then p(4) is 4² - 8*4 + 5 = 16 - 32 + 5 = (16 + 5) - 32 = 21 - 32 = -11. Therefore, p(3)/p(4) is (-10)/(-11) = 10/11. Yep, same result. So that seems correct.

But let me think again if there's another way this could be approached. For example, maybe using Vieta's formula? Let me try that. If the roots are 4 - sqrt(11) and 4 + sqrt(11), then the sum of the roots is (4 - sqrt(11)) + (4 + sqrt(11)) = 8. The product of the roots is (4 - sqrt(11))(4 + sqrt(11)) = 16 - (sqrt(11))² = 16 - 11 = 5. Therefore, the quadratic polynomial can be written as x² - (sum)x + product = x² - 8x + 5, which matches what I found earlier. So that's consistent.

Alternatively, if I didn't remember the conjugate root theorem, I could have tried to construct the polynomial. Let me try that approach. Suppose that 4 - sqrt(11) is a root of p(x). Then p(4 - sqrt(11)) = 0. Let me assume p(x) is quadratic, so p(x) = ax² + bx + c, with a, b, c integers. Then plugging in x = 4 - sqrt(11):

a(4 - sqrt(11))² + b(4 - sqrt(11)) + c = 0.

Let me compute (4 - sqrt(11))². That is 16 - 8 sqrt(11) + 11 = 27 - 8 sqrt(11). Therefore, substituting back:

a(27 - 8 sqrt(11)) + b(4 - sqrt(11)) + c = 0.

Expanding this:

27a - 8a sqrt(11) + 4b - b sqrt(11) + c = 0.

Grouping the rational and irrational parts:

(27a + 4b + c) + (-8a - b) sqrt(11) = 0.

Since sqrt(11) is irrational, the coefficients of the rational and irrational parts must both be zero. Therefore, we have the system of equations:

27a + 4b + c = 0,

-8a - b = 0.

From the second equation: -8a - b = 0 => b = -8a.

Substituting into the first equation: 27a + 4(-8a) + c = 0 => 27a - 32a + c = 0 => -5a + c = 0 => c = 5a.

Therefore, the polynomial is p(x) = ax² + bx + c = a x² - 8a x + 5a = a(x² - 8x + 5). Which is the same as before. Therefore, regardless of the value of a (as long as a ≠ 0), the polynomial is a multiple of x² - 8x + 5, so the ratio p(3)/p(4) is 10/11 as before.

Therefore, regardless of the approach, the answer is 10/11. So I think that's solid.

But just to make sure, let me check if there's any other possible polynomial. Suppose someone thought that maybe there's another root? But since it's quadratic, there are only two roots. And since the coefficients are integers, the conjugate must be there. So there's no other possibility. Therefore, the minimal polynomial is indeed x² - 8x + 5, and any other polynomial with integer coefficients would be a multiple of this. Therefore, the ratio is fixed.

Alternatively, if someone tried to use another method, like plugging in values? But since we need p(3)/p(4), and we have the polynomial, it's straightforward. But maybe if we didn't know the polynomial, how would we proceed?

Alternatively, suppose we didn't know the conjugate root theorem. Let me think. If we have a quadratic with integer coefficients, and one root is 4 - sqrt(11), then we can write the polynomial as (x - (4 - sqrt(11)))(x - r), where r is the other root. Then expanding this, we get x² - (4 - sqrt(11) + r)x + (4 - sqrt(11))r. Since coefficients must be integers, the coefficients of x and the constant term must be integers. Let me denote the coefficients:

Coefficient of x: -(4 - sqrt(11) + r) must be integer. Therefore, 4 - sqrt(11) + r must be rational. Let me write that as r + 4 - sqrt(11) is rational. Therefore, since sqrt(11) is irrational, the only way this sum is rational is if the irrational parts cancel out. Therefore, r must be of the form 4 + sqrt(11), so that when added to 4 - sqrt(11), the sqrt(11) terms cancel. Therefore, r = 4 + sqrt(11). Therefore, the other root is 4 + sqrt(11), which brings us back to the same conclusion. Therefore, the polynomial is (x - (4 - sqrt(11)))(x - (4 + sqrt(11))) = x² - 8x + 5. So that's another way to see it.

Therefore, regardless of the approach, the polynomial is x² - 8x + 5, and the ratio p(3)/p(4) is 10/11. Therefore, I think that's the answer.

But just to be thorough, let me check if there's a possibility that the polynomial could have a different form. For example, suppose someone thought that maybe the polynomial is monic? But the problem doesn't specify that. However, even if it's not monic, as we saw earlier, the ratio remains the same. So even if the polynomial was 2x² - 16x + 10, then p(3) would be 2*(9) -16*(3) +10 = 18 - 48 +10 = -20, and p(4) would be 2*(16) -16*(4) +10 = 32 - 64 +10 = -22. Then p(3)/p(4) is (-20)/(-22) = 10/11. Same result. So even if the polynomial is scaled, the ratio remains the same. Therefore, regardless of the leading coefficient, the ratio is 10/11.

Therefore, I think that's solid. So the answer is 10/11.

**Final Answer**
\boxed{\dfrac{10}{11}}

To solve the problem, we are given that $ p(x) $ is a **quadratic polynomial with integer coefficients** and that one of its roots is $ 4 - \sqrt{11} $. We are asked to compute the value of $ \frac{p(3)}{p(4)} $.

---

### Step 1: Use the Conjugate Root Theorem

Since $ p(x) $ has **integer coefficients** and one of its roots is $ 4 - \sqrt{11} $, the **conjugate root** $ 4 + \sqrt{11} $ must also be a root. This is a consequence of the **Conjugate Root Theorem**, which states that irrational roots of polynomials with rational (or integer) coefficients occur in conjugate pairs.

Thus, the two roots of the polynomial are:
$$
r_1 = 4 - \sqrt{11}, \quad r_2 = 4 + \sqrt{11}
$$

---

### Step 2: Construct the Polynomial

A quadratic polynomial with these roots can be written as:
$$
p(x) = (x - (4 - \sqrt{11}))(x - (4 + \sqrt{11}))
$$

This is a difference of squares:
$$
p(x) = (x - 4 + \sqrt{11})(x - 4 - \sqrt{11}) = (x - 4)^2 - (\sqrt{11})^2
$$

Now compute:
$$
(x - 4)^2 = x^2 - 8x + 16, \quad (\sqrt{11})^2 = 11
$$
$$
p(x) = x^2 - 8x + 16 - 11 = x^2 - 8x + 5
$$

So, the polynomial is:
$$
p(x) = x^2 - 8x + 5
$$

---

### Step 3: Evaluate $ p(3) $ and $ p(4) $

Now compute:
$$
p(3) = 3^2 - 8 \cdot 3 + 5 = 9 - 24 + 5 = -10
$$
$$
p(4) = 4^2 - 8 \cdot 4 + 5 = 16 - 32 + 5 = -11
$$

Thus:
$$
\frac{p(3)}{p(4)} = \frac{-10}{-11} = \frac{10}{11}
$$

---

### Step 4: Consider General Form of the Polynomial

Note that any scalar multiple of this polynomial will also have integer coefficients. That is, for any integer $ k \neq 0 $, the polynomial $ k(x^2 - 8x + 5) $ also satisfies the given conditions.

Let’s verify that the ratio $ \frac{p(3)}{p(4)} $ remains the same:
$$
p(x) = k(x^2 - 8x + 5) \Rightarrow p(3) = k(-10), \quad p(4) = k(-11)
$$
$$
\frac{p(3)}{p(4)} = \frac{k(-10)}{k(-11)} = \frac{10}{11}
$$

So, the ratio is **independent of the scalar multiple**.

---

### Final Answer

$$
\boxed{\dfrac{10}{11}}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of p(3)/p(4) where p(x) is a quadratic polynomial with integer coefficients and one of its roots is 4 - sqrt(11). Hmm, let me think about how to approach this.

First, since p(x) is a quadratic polynomial with integer coefficients and one of its roots is 4 - sqrt(11), I remember that irrational roots of polynomials with integer coefficients come in conjugate pairs. That means if 4 - sqrt(11) is a root, then its conjugate, which would be 4 + sqrt(11), must also be a root. Is that right? Let me confirm. Yes, because if you have a polynomial with integer coefficients and a root that's a radical like a + sqrt(b), then the conjugate a - sqrt(b) must also be a root to ensure that when you multiply out the factors, the irrational parts cancel out, leaving integer coefficients. So, that makes sense.

Therefore, the quadratic polynomial p(x) can be written as p(x) = (x - (4 - sqrt(11)))(x - (4 + sqrt(11))). Let me check that. If I expand this, it should give me a quadratic with integer coefficients. Let me do the expansion.

First, multiply the two binomials:

(x - 4 + sqrt(11))(x - 4 - sqrt(11)). Wait, actually, the roots are 4 - sqrt(11) and 4 + sqrt(11), so the factors are (x - (4 - sqrt(11))) and (x - (4 + sqrt(11))). Let me write that as (x - 4 + sqrt(11))(x - 4 - sqrt(11)). That looks like a difference of squares. Let me set a = x - 4 and b = sqrt(11). Then the expression becomes (a + b)(a - b) = a² - b². So substituting back, that would be (x - 4)² - (sqrt(11))². Let me compute that.

(x - 4)² is x² - 8x + 16, and (sqrt(11))² is 11. Therefore, the polynomial is x² - 8x + 16 - 11 = x² - 8x + 5. So p(x) = x² - 8x + 5. Wait, but the problem says that p(x) has integer coefficients, which this does. So that seems correct.

But hold on, is this the only possible polynomial? Because any scalar multiple of this polynomial would also have integer coefficients. For example, if we multiply by 2, we get 2x² - 16x + 10, which also has integer coefficients. However, the problem says "a quadratic polynomial with integer coefficients", so there could be multiple polynomials. But since we are asked to compute p(3)/p(4), maybe the ratio is the same regardless of the scalar multiple? Let me check.

Suppose p(x) = k(x² - 8x + 5) where k is an integer. Then p(3) = k(9 - 24 + 5) = k(-10), and p(4) = k(16 - 32 + 5) = k(-11). Therefore, p(3)/p(4) = (-10k)/(-11k) = 10/11. So regardless of the value of k (as long as k ≠ 0, which it can't be because then p(x) would be the zero polynomial, which isn't quadratic), the ratio is 10/11. Therefore, the answer is 10/11. Wait, so even though there are multiple polynomials, the ratio ends up being the same? That seems right because the k cancels out. Therefore, the answer is 10/11.

But let me verify this by actually computing p(3) and p(4) for the specific polynomial I found. Let me take p(x) = x² - 8x + 5. Then p(3) is 3² - 8*3 + 5 = 9 - 24 + 5 = (9 + 5) - 24 = 14 - 24 = -10. Then p(4) is 4² - 8*4 + 5 = 16 - 32 + 5 = (16 + 5) - 32 = 21 - 32 = -11. Therefore, p(3)/p(4) is (-10)/(-11) = 10/11. Yep, same result. So that seems correct.

But let me think again if there's another way this could be approached. For example, maybe using Vieta's formula? Let me try that. If the roots are 4 - sqrt(11) and 4 + sqrt(11), then the sum of the roots is (4 - sqrt(11)) + (4 + sqrt(11)) = 8. The product of the roots is (4 - sqrt(11))(4 + sqrt(11)) = 16 - (sqrt(11))² = 16 - 11 = 5. Therefore, the quadratic polynomial can be written as x² - (sum)x + product = x² - 8x + 5, which matches what I found earlier. So that's consistent.

Alternatively, if I didn't remember the conjugate root theorem, I could have tried to construct the polynomial. Let me try that approach. Suppose that 4 - sqrt(11) is a root of p(x). Then p(4 - sqrt(11)) = 0. Let me assume p(x) is quadratic, so p(x) = ax² + bx + c, with a, b, c integers. Then plugging in x = 4 - sqrt(11):

a(4 - sqrt(11))² + b(4 - sqrt(11)) + c = 0.

Let me compute (4 - sqrt(11))². That is 16 - 8 sqrt(11) + 11 = 27 - 8 sqrt(11). Therefore, substituting back:

a(27 - 8 sqrt(11)) + b(4 - sqrt(11)) + c = 0.

Expanding this:

27a - 8a sqrt(11) + 4b - b sqrt(11) + c = 0.

Grouping the rational and irrational parts:

(27a + 4b + c) + (-8a - b) sqrt(11) = 0.

Since sqrt(11) is irrational, the coefficients of the rational and irrational parts must both be zero. Therefore, we have the system of equations:

27a + 4b + c = 0,

-8a - b = 0.

From the second equation: -8a - b = 0 => b = -8a.

Substituting into the first equation: 27a + 4(-8a) + c = 0 => 27a - 32a + c = 0 => -5a + c = 0 => c = 5a.

Therefore, the polynomial is p(x) = ax² + bx + c = a x² - 8a x + 5a = a(x² - 8x + 5). Which is the same as before. Therefore, regardless of the value of a (as long as a ≠ 0), the polynomial is a multiple of x² - 8x + 5, so the ratio p(3)/p(4) is 10/11 as before.

Therefore, regardless of the approach, the answer is 10/11. So I think that's solid.

But just to make sure, let me check if there's any other possible polynomial. Suppose someone thought that maybe there's another root? But since it's quadratic, there are only two roots. And since the coefficients are integers, the conjugate must be there. So there's no other possibility. Therefore, the minimal polynomial is indeed x² - 8x + 5, and any other polynomial with integer coefficients would be a multiple of this. Therefore, the ratio is fixed.

Alternatively, if someone tried to use another method, like plugging in values? But since we need p(3)/p(4), and we have the polynomial, it's straightforward. But maybe if we didn't know the polynomial, how would we proceed?

Alternatively, suppose we didn't know the conjugate root theorem. Let me think. If we have a quadratic with integer coefficients, and one root is 4 - sqrt(11), then we can write the polynomial as (x - (4 - sqrt(11)))(x - r), where r is the other root. Then expanding this, we get x² - (4 - sqrt(11) + r)x + (4 - sqrt(11))r. Since coefficients must be integers, the coefficients of x and the constant term must be integers. Let me denote the coefficients:

Coefficient of x: -(4 - sqrt(11) + r) must be integer. Therefore, 4 - sqrt(11) + r must be rational. Let me write that as r + 4 - sqrt(11) is rational. Therefore, since sqrt(11) is irrational, the only way this sum is rational is if the irrational parts cancel out. Therefore, r must be of the form 4 + sqrt(11), so that when added to 4 - sqrt(11), the sqrt(11) terms cancel. Therefore, r = 4 + sqrt(11). Therefore, the other root is 4 + sqrt(11), which brings us back to the same conclusion. Therefore, the polynomial is (x - (4 - sqrt(11)))(x - (4 + sqrt(11))) = x² - 8x + 5. So that's another way to see it.

Therefore, regardless of the approach, the polynomial is x² - 8x + 5, and the ratio p(3)/p(4) is 10/11. Therefore, I think that's the answer.

But just to be thorough, let me check if there's a possibility that the polynomial could have a different form. For example, suppose someone thought that maybe the polynomial is monic? But the problem doesn't specify that. However, even if it's not monic, as we saw earlier, the ratio remains the same. So even if the polynomial was 2x² - 16x + 10, then p(3) would be 2*(9) -16*(3) +10 = 18 - 48 +10 = -20, and p(4) would be 2*(16) -16*(4) +10 = 32 - 64 +10 = -22. Then p(3)/p(4) is (-20)/(-22) = 10/11. Same result. So even if the polynomial is scaled, the ratio remains the same. Therefore, regardless of the leading coefficient, the ratio is 10/11.

Therefore, I think that's solid. So the answer is 10/11.

**Final Answer**
\boxed{\dfrac{10}{11}}

To solve the problem, we are given that $ p(x) $ is a **quadratic polynomial with integer coefficients** and that one of its roots is $ 4 - \sqrt{11} $. We are asked to compute the value of $ \frac{p(3)}{p(4)} $.

---

### Step 1: Use the Conjugate Root Theorem

Since $ p(x) $ has **integer coefficients** and one of its roots is $ 4 - \sqrt{11} $, the **conjugate root** $ 4 + \sqrt{11} $ must also be a root. This is a consequence of the **Conjugate Root Theorem**, which states that irrational roots of polynomials with rational (or integer) coefficients occur in conjugate pairs.

Thus, the two roots of the polynomial are:
$$
r_1 = 4 - \sqrt{11}, \quad r_2 = 4 + \sqrt{11}
$$

---

### Step 2: Construct the Polynomial

A quadratic polynomial with these roots can be written as:
$$
p(x) = (x - (4 - \sqrt{11}))(x - (4 + \sqrt{11}))
$$

This is a difference of squares:
$$
p(x) = (x - 4 + \sqrt{11})(x - 4 - \sqrt{11}) = (x - 4)^2 - (\sqrt{11})^2
$$

Now compute:
$$
(x - 4)^2 = x^2 - 8x + 16, \quad (\sqrt{11})^2 = 11
$$
$$
p(x) = x^2 - 8x + 16 - 11 = x^2 - 8x + 5
$$

So, the polynomial is:
$$
p(x) = x^2 - 8x + 5
$$

---

### Step 3: Evaluate $ p(3) $ and $ p(4) $

Now compute:
$$
p(3) = 3^2 - 8 \cdot 3 + 5 = 9 - 24 + 5 = -10
$$
$$
p(4) = 4^2 - 8 \cdot 4 + 5 = 16 - 32 + 5 = -11
$$

Thus:
$$
\frac{p(3)}{p(4)} = \frac{-10}{-11} = \frac{10}{11}
$$

---

### Step 4: Consider General Form of the Polynomial

Note that any scalar multiple of this polynomial will also have integer coefficients. That is, for any integer $ k \neq 0 $, the polynomial $ k(x^2 - 8x + 5) $ also satisfies the given conditions.

Let’s verify that the ratio $ \frac{p(3)}{p(4)} $ remains the same:
$$
p(x) = k(x^2 - 8x + 5) \Rightarrow p(3) = k(-10), \quad p(4) = k(-11)
$$
$$
\frac{p(3)}{p(4)} = \frac{k(-10)}{k(-11)} = \frac{10}{11}
$$

So, the ratio is **independent of the scalar multiple**.

---

### Final Answer

$$
\boxed{\dfrac{10}{11}}
$$

Solution: Because $p(x)$ has integer coefficients (in particular, because it has rational coefficients), the other root of $p(x)$ must be the radical conjugate of $4-\sqrt{11},$ which is $4+\sqrt{11}.$ Then, $p(x)$ must take the form \[p(x) = A(x-(4-\sqrt{11}))(x-(4+\sqrt{11}))\]for some nonzero constant $A$. This means that \[p(3) = A(-1+\sqrt{11})(-1-\sqrt{11}) = -10A\]and \[p(4) = A(\sqrt{11})(-\sqrt{11}) = -11A,\]so \[\frac{p(3)}{p(4)} = \frac{-10A}{-11A} = \boxed{\frac{10}{11}}.\]


Explanation:
Okay, so I need to figure out what fraction of 2 feet is 3 inches. Hmm, let me start by understanding the problem. It's asking for the fraction that represents 3 inches in relation to 2 feet. So basically, I need to compare 3 inches to 2 feet and express that comparison as a fraction.

First, I should make sure both measurements are in the same unit. Right now, one is in feet and the other is in inches. I know that 1 foot is equal to 12 inches. So maybe I should convert 2 feet into inches to make the comparison easier. Let me do that.

If 1 foot is 12 inches, then 2 feet would be 2 times 12 inches. Let me calculate that: 2 * 12 = 24 inches. So, 2 feet is equal to 24 inches.

Now, the problem is asking what fraction of 24 inches is 3 inches. So, in other words, how many times does 3 inches fit into 24 inches? But since we want a fraction, it's just 3 divided by 24.

Let me write that down: 3 inches / 24 inches. The inches unit cancels out, which makes sense because we're comparing two lengths. So that simplifies to 3/24.

Now, I need to simplify this fraction. Both numerator and denominator are divisible by 3. Let me divide numerator and denominator by 3.

3 divided by 3 is 1, and 24 divided by 3 is 8. So, 3/24 simplifies to 1/8.

Wait, let me check that again. If I have 3 inches over 24 inches, then yes, dividing both by 3 gives 1/8. So, 3 inches is 1/8 of 2 feet.

But let me verify this another way to make sure I didn't make a mistake. Let's think about it in terms of fractions without converting units.

Original question: What fraction of 2 feet is 3 inches?

Alternatively, I can convert 3 inches to feet first. Since 1 foot is 12 inches, 3 inches is 3/12 feet. Simplifying that, 3/12 is 1/4 foot.

So, now the question is, what fraction of 2 feet is 1/4 foot? That would be (1/4) divided by 2.

Dividing by 2 is the same as multiplying by 1/2, so (1/4) * (1/2) = 1/8.

Oh, same answer! So that confirms that 3 inches is 1/8 of 2 feet.

Therefore, the fraction is 1/8.

Wait, just to make sure I didn't mix up anything. Let me think again.

If 2 feet is 24 inches, then 3 inches is how much? Let me divide 24 by 3. 24 divided by 3 is 8. So, 3 inches is 1/8 of 24 inches. Yep, that's the same as before.

Alternatively, if I think about proportions. Let me set up a proportion.

Let me let x be the fraction we're trying to find. So, x * 2 feet = 3 inches.

But to solve for x, we need to have the same units. So, converting 3 inches to feet, as before, is 3/12 feet, which is 1/4 feet.

So, x * 2 feet = 1/4 feet.

Divide both sides by 2 feet: x = (1/4) / 2 = 1/8.

Same result again.

So, all methods lead to 1/8. Therefore, I think that's the correct answer.

But just to make sure, let me think of another approach.

Suppose I consider the ratio of 3 inches to 2 feet. Since ratios can be expressed as fractions, so 3 inches : 2 feet.

To express this ratio as a fraction, we need to have the same units. So, converting 2 feet to inches is 24 inches, so the ratio is 3 inches : 24 inches, which is 3/24, simplifies to 1/8.

Same answer again.

Alternatively, if I convert 3 inches to feet, which is 0.25 feet, and then divide by 2 feet, that's 0.25 / 2 = 0.125, which is 1/8.

Yes, decimal conversion also gives me 1/8.

So, all different methods of approaching the problem lead me to the same answer, 1/8.

Therefore, I can be confident that the fraction of 2 feet that is 3 inches is 1/8.

**Final Answer**
The fraction of 2 feet that is 3 inches is \boxed{\dfrac{1}{8}}.

To determine what fraction of 2 feet is 3 inches, we need to express both measurements in the same unit and then compute the ratio.

---

### Step 1: Convert Units

We are given:
- 2 feet
- 3 inches

Since the units are different, we convert 2 feet to inches. Recall that:

$$
1 \text{ foot} = 12 \text{ inches}
$$

So,

$$
2 \text{ feet} = 2 \times 12 = 24 \text{ inches}
$$

Now, both measurements are in inches:
- 2 feet = 24 inches
- 3 inches = 3 inches

---

### Step 2: Compute the Fraction

We want to find what fraction of 24 inches is 3 inches. This is:

$$
\frac{3}{24}
$$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3:

$$
\frac{3}{24} = \frac{1}{8}
$$

---

### Step 3: Final Answer

Thus, 3 inches is $\frac{1}{8}$ of 2 feet.

$$
\boxed{\dfrac{1}{8}}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what fraction of 2 feet is 3 inches. Hmm, let me start by understanding the problem. It's asking for the fraction that represents 3 inches in relation to 2 feet. So basically, I need to compare 3 inches to 2 feet and express that comparison as a fraction.

First, I should make sure both measurements are in the same unit. Right now, one is in feet and the other is in inches. I know that 1 foot is equal to 12 inches. So maybe I should convert 2 feet into inches to make the comparison easier. Let me do that.

If 1 foot is 12 inches, then 2 feet would be 2 times 12 inches. Let me calculate that: 2 * 12 = 24 inches. So, 2 feet is equal to 24 inches.

Now, the problem is asking what fraction of 24 inches is 3 inches. So, in other words, how many times does 3 inches fit into 24 inches? But since we want a fraction, it's just 3 divided by 24.

Let me write that down: 3 inches / 24 inches. The inches unit cancels out, which makes sense because we're comparing two lengths. So that simplifies to 3/24.

Now, I need to simplify this fraction. Both numerator and denominator are divisible by 3. Let me divide numerator and denominator by 3.

3 divided by 3 is 1, and 24 divided by 3 is 8. So, 3/24 simplifies to 1/8.

Wait, let me check that again. If I have 3 inches over 24 inches, then yes, dividing both by 3 gives 1/8. So, 3 inches is 1/8 of 2 feet.

But let me verify this another way to make sure I didn't make a mistake. Let's think about it in terms of fractions without converting units.

Original question: What fraction of 2 feet is 3 inches?

Alternatively, I can convert 3 inches to feet first. Since 1 foot is 12 inches, 3 inches is 3/12 feet. Simplifying that, 3/12 is 1/4 foot.

So, now the question is, what fraction of 2 feet is 1/4 foot? That would be (1/4) divided by 2.

Dividing by 2 is the same as multiplying by 1/2, so (1/4) * (1/2) = 1/8.

Oh, same answer! So that confirms that 3 inches is 1/8 of 2 feet.

Therefore, the fraction is 1/8.

Wait, just to make sure I didn't mix up anything. Let me think again.

If 2 feet is 24 inches, then 3 inches is how much? Let me divide 24 by 3. 24 divided by 3 is 8. So, 3 inches is 1/8 of 24 inches. Yep, that's the same as before.

Alternatively, if I think about proportions. Let me set up a proportion.

Let me let x be the fraction we're trying to find. So, x * 2 feet = 3 inches.

But to solve for x, we need to have the same units. So, converting 3 inches to feet, as before, is 3/12 feet, which is 1/4 feet.

So, x * 2 feet = 1/4 feet.

Divide both sides by 2 feet: x = (1/4) / 2 = 1/8.

Same result again.

So, all methods lead to 1/8. Therefore, I think that's the correct answer.

But just to make sure, let me think of another approach.

Suppose I consider the ratio of 3 inches to 2 feet. Since ratios can be expressed as fractions, so 3 inches : 2 feet.

To express this ratio as a fraction, we need to have the same units. So, converting 2 feet to inches is 24 inches, so the ratio is 3 inches : 24 inches, which is 3/24, simplifies to 1/8.

Same answer again.

Alternatively, if I convert 3 inches to feet, which is 0.25 feet, and then divide by 2 feet, that's 0.25 / 2 = 0.125, which is 1/8.

Yes, decimal conversion also gives me 1/8.

So, all different methods of approaching the problem lead me to the same answer, 1/8.

Therefore, I can be confident that the fraction of 2 feet that is 3 inches is 1/8.

**Final Answer**
The fraction of 2 feet that is 3 inches is \boxed{\dfrac{1}{8}}.

To determine what fraction of 2 feet is 3 inches, we need to express both measurements in the same unit and then compute the ratio.

---

### Step 1: Convert Units

We are given:
- 2 feet
- 3 inches

Since the units are different, we convert 2 feet to inches. Recall that:

$$
1 \text{ foot} = 12 \text{ inches}
$$

So,

$$
2 \text{ feet} = 2 \times 12 = 24 \text{ inches}
$$

Now, both measurements are in inches:
- 2 feet = 24 inches
- 3 inches = 3 inches

---

### Step 2: Compute the Fraction

We want to find what fraction of 24 inches is 3 inches. This is:

$$
\frac{3}{24}
$$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3:

$$
\frac{3}{24} = \frac{1}{8}
$$

---

### Step 3: Final Answer

Thus, 3 inches is $\frac{1}{8}$ of 2 feet.

$$
\boxed{\dfrac{1}{8}}
$$

Solution: Since 2 feet is $2\cdot12=24$ inches, 3 inches is $3/24=\boxed{\frac{1}{8}}$ of 2 feet.


Explanation:
Okay, so I need to simplify this expression: (1/5) multiplied by (8/7) divided by (12/20). Let me think about how to approach this.

First, I remember that dividing by a fraction is the same as multiplying by its reciprocal. So maybe I can rewrite the division part as a multiplication. Let me write that down.

The original problem is (1/5) * (8/7) ÷ (12/20). So, dividing by (12/20) is the same as multiplying by (20/12). Let me check that. Yes, because dividing by a fraction flips the numerator and denominator. So, ÷ (a/b) becomes * (b/a).

So, rewriting the expression, it becomes (1/5) * (8/7) * (20/12). Now, I need to multiply these three fractions together. To multiply fractions, I multiply the numerators together and the denominators together.

Let me write that out. Numerators: 1 * 8 * 20. Denominators: 5 * 7 * 12. Let me compute each part step by step.

First, the numerators: 1 times 8 is 8, and then 8 times 20. Let me calculate 8*20. That's 160. So the numerator is 160.

Now the denominators: 5 times 7 is 35, and then 35 times 12. Let me compute that. 35*12. Hmm, 35*10 is 350, and 35*2 is 70, so 350 + 70 is 420. So the denominator is 420.

So now the fraction is 160/420. Now I need to simplify this fraction. Let me see if I can divide numerator and denominator by a common factor.

First, let me check if both numbers are divisible by 10. 160 ÷ 10 is 16, and 420 ÷ 10 is 42. So that simplifies to 16/42.

Wait, can I simplify that further? Let me check. 16 and 42. Let's see, 16 factors are 2^4, and 42 factors are 2*3*7. So they have a common factor of 2. Dividing numerator and denominator by 2: 16 ÷ 2 is 8, and 42 ÷ 2 is 21. So now it's 8/21.

Is 8 and 21 have any common factors? 8 is 2^3, and 21 is 3*7. No common factors, so 8/21 is the simplified form.

Wait, but let me check my steps again to make sure I didn't make a mistake.

Original problem: (1/5)*(8/7)/(12/20). Convert division to multiplication by reciprocal: (1/5)*(8/7)*(20/12). Multiply numerators: 1*8*20 = 160. Multiply denominators: 5*7*12 = 420. Simplify 160/420. Divided by 10: 16/42. Divided by 2: 8/21. That seems right.

Alternatively, maybe I can simplify before multiplying. Let me try that approach to cross-verify.

So, when multiplying fractions, it's often easier to cancel common factors before multiplying. Let me try that.

Original expression after converting division to multiplication: (1/5) * (8/7) * (20/12).

Let me write all numerators and denominators:

Numerators: 1, 8, 20

Denominators: 5, 7, 12

So, let me see if any numerator and denominator can be simplified.

Looking at 1 and 5: 1 is numerator, 5 is denominator. No common factors except 1.

Looking at 8 and 5: 8 and 5 have no common factors.

Looking at 8 and 7: no common factors.

Looking at 8 and 12: 8 and 12 have a common factor of 4.

Looking at 20 and 5: 20 and 5 have a common factor of 5.

Looking at 20 and 7: no common factors.

Looking at 20 and 12: 20 and 12 have a common factor of 4.

So maybe I can cancel these. Let me try.

First, let me note that 20 in the numerator and 5 in the denominator. 20 divided by 5 is 4, and 5 divided by 5 is 1. So, if I cancel 20 and 5, that would be replacing 20 with 4 and 5 with 1.

Similarly, 8 and 12 have a common factor of 4. Let me see. If I divide 8 by 4, it becomes 2, and divide 12 by 4, it becomes 3.

So let me do that step by step.

First, let me write the fractions again:

(1/5) * (8/7) * (20/12)

Let me rearrange the numerators and denominators to see common factors:

Numerators: 1, 8, 20

Denominators: 5, 7, 12

So, let me write them as:

(1 * 8 * 20) / (5 * 7 * 12)

Now, let me factor each number into primes:

1 is 1.

8 is 2^3.

20 is 2^2 * 5.

Denominator:

5 is 5.

7 is 7.

12 is 2^2 * 3.

So, combining all numerators:

Numerator factors: 1 * 2^3 * 2^2 * 5 = 2^(3+2) * 5 = 2^5 * 5

Denominator factors: 5 * 7 * 2^2 * 3 = 2^2 * 3 * 5 * 7

So, the entire fraction is (2^5 * 5) / (2^2 * 3 * 5 * 7)

Now, let's cancel common factors.

First, 2^5 divided by 2^2 is 2^(5-2) = 2^3.

Then, 5 in the numerator and 5 in the denominator cancel out.

So, remaining factors in numerator: 2^3

Remaining factors in denominator: 3 * 7

Therefore, the simplified fraction is 2^3 / (3 * 7) = 8 / 21.

So that's the same result as before. So that's reassuring.

Alternatively, maybe I can do step by step simplification. Let me try that.

Original expression: (1/5) * (8/7) * (20/12)

First, let me multiply (1/5) and (8/7).

Multiplying those two: (1*8)/(5*7) = 8/35.

Then, multiply that by (20/12). So, 8/35 * 20/12.

Again, let me simplify before multiplying.

Looking at 8 and 12: common factor of 4. 8 ÷ 4 = 2, 12 ÷ 4 = 3.

Looking at 20 and 35: common factor of 5. 20 ÷ 5 = 4, 35 ÷ 5 = 7.

So, simplifying:

(8/35) * (20/12) = (2/7) * (4/3) = (2*4)/(7*3) = 8/21.

Same result.

So, regardless of the method, I get 8/21.

Wait, let me check once more. Maybe I can compute decimal values to verify.

Original expression: (1/5) * (8/7) ÷ (12/20).

Compute each part:

1/5 is 0.2

8/7 is approximately 1.142857...

12/20 is 0.6, so dividing by 0.6 is the same as multiplying by 1/0.6 ≈ 1.666666...

So, let me compute step by step:

First, 0.2 * 1.142857 ≈ 0.2285714

Then, divide by 0.6: 0.2285714 / 0.6 ≈ 0.380952...

Now, 8/21 is approximately 0.380952... which matches.

So that's correct.

Alternatively, compute 8 divided by 21. 21*0.38 is 7.98, so yes, approximately 0.38095.

Therefore, the simplified fraction is 8/21.

So, I think that's the correct answer.

But just to recap all steps once again to make sure I didn't skip anything:

Original problem: (1/5) * (8/7) ÷ (12/20)

Convert division to multiplication by reciprocal: (1/5) * (8/7) * (20/12)

Multiply numerators: 1*8*20 = 160

Multiply denominators: 5*7*12 = 420

Simplify 160/420 by dividing numerator and denominator by 10: 16/42

Then divide by 2: 8/21.

Yes, that's correct.

Alternatively, simplifying before multiplying:

(1/5)*(8/7)*(20/12) = (1/5)*(8/7)*(5/3) [Wait, 20/12 simplifies to 5/3? Wait, 20 divided by 4 is 5, 12 divided by 4 is 3. So yes, 20/12 = 5/3.

So, substituting that in, the expression becomes (1/5) * (8/7) * (5/3).

Now, notice that 1/5 and 5/3 have a common factor of 5.

So, 1/5 * 5/3 = (1*5)/(5*3) = 5/15 = 1/3.

Then, multiply by 8/7: 1/3 * 8/7 = 8/21.

Oh, that's another way to simplify. So that's even quicker.

So, that's another method.

So, that's even better. So, if I simplify 20/12 first to 5/3, then the 5 in the denominator cancels with the 5 in the numerator from 1/5 * 5/3.

So, that's a more efficient way.

So, that's why it's important to simplify before multiplying.

So, in conclusion, regardless of the method, the answer is 8/21.

Therefore, the simplified form of the given expression is 8/21.

**Final Answer**
The simplified expression is \boxed{\dfrac{8}{21}}.

To simplify the expression:

$$
\frac{1}{5} \cdot \frac{8}{7} \div \frac{12}{20}
$$

---

### Step 1: Convert Division to Multiplication

Recall that dividing by a fraction is the same as multiplying by its reciprocal. So:

$$
\frac{1}{5} \cdot \frac{8}{7} \div \frac{12}{20} = \frac{1}{5} \cdot \frac{8}{7} \cdot \frac{20}{12}
$$

---

### Step 2: Multiply the Fractions

Now, multiply the numerators and denominators:

- Numerators: $1 \cdot 8 \cdot 20 = 160$
- Denominators: $5 \cdot 7 \cdot 12 = 420$

So the expression becomes:

$$
\frac{160}{420}
$$

---

### Step 3: Simplify the Fraction

To simplify $\frac{160}{420}$, find the greatest common divisor (GCD) of 160 and 420. The GCD is 20.

$$
\frac{160 \div 20}{420 \div 20} = \frac{8}{21}
$$

---

### Step 4: Final Answer

The simplified form of the expression is:

$$
\boxed{\dfrac{8}{21}}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: (1/5) multiplied by (8/7) divided by (12/20). Let me think about how to approach this.

First, I remember that dividing by a fraction is the same as multiplying by its reciprocal. So maybe I can rewrite the division part as a multiplication. Let me write that down.

The original problem is (1/5) * (8/7) ÷ (12/20). So, dividing by (12/20) is the same as multiplying by (20/12). Let me check that. Yes, because dividing by a fraction flips the numerator and denominator. So, ÷ (a/b) becomes * (b/a).

So, rewriting the expression, it becomes (1/5) * (8/7) * (20/12). Now, I need to multiply these three fractions together. To multiply fractions, I multiply the numerators together and the denominators together.

Let me write that out. Numerators: 1 * 8 * 20. Denominators: 5 * 7 * 12. Let me compute each part step by step.

First, the numerators: 1 times 8 is 8, and then 8 times 20. Let me calculate 8*20. That's 160. So the numerator is 160.

Now the denominators: 5 times 7 is 35, and then 35 times 12. Let me compute that. 35*12. Hmm, 35*10 is 350, and 35*2 is 70, so 350 + 70 is 420. So the denominator is 420.

So now the fraction is 160/420. Now I need to simplify this fraction. Let me see if I can divide numerator and denominator by a common factor.

First, let me check if both numbers are divisible by 10. 160 ÷ 10 is 16, and 420 ÷ 10 is 42. So that simplifies to 16/42.

Wait, can I simplify that further? Let me check. 16 and 42. Let's see, 16 factors are 2^4, and 42 factors are 2*3*7. So they have a common factor of 2. Dividing numerator and denominator by 2: 16 ÷ 2 is 8, and 42 ÷ 2 is 21. So now it's 8/21.

Is 8 and 21 have any common factors? 8 is 2^3, and 21 is 3*7. No common factors, so 8/21 is the simplified form.

Wait, but let me check my steps again to make sure I didn't make a mistake.

Original problem: (1/5)*(8/7)/(12/20). Convert division to multiplication by reciprocal: (1/5)*(8/7)*(20/12). Multiply numerators: 1*8*20 = 160. Multiply denominators: 5*7*12 = 420. Simplify 160/420. Divided by 10: 16/42. Divided by 2: 8/21. That seems right.

Alternatively, maybe I can simplify before multiplying. Let me try that approach to cross-verify.

So, when multiplying fractions, it's often easier to cancel common factors before multiplying. Let me try that.

Original expression after converting division to multiplication: (1/5) * (8/7) * (20/12).

Let me write all numerators and denominators:

Numerators: 1, 8, 20

Denominators: 5, 7, 12

So, let me see if any numerator and denominator can be simplified.

Looking at 1 and 5: 1 is numerator, 5 is denominator. No common factors except 1.

Looking at 8 and 5: 8 and 5 have no common factors.

Looking at 8 and 7: no common factors.

Looking at 8 and 12: 8 and 12 have a common factor of 4.

Looking at 20 and 5: 20 and 5 have a common factor of 5.

Looking at 20 and 7: no common factors.

Looking at 20 and 12: 20 and 12 have a common factor of 4.

So maybe I can cancel these. Let me try.

First, let me note that 20 in the numerator and 5 in the denominator. 20 divided by 5 is 4, and 5 divided by 5 is 1. So, if I cancel 20 and 5, that would be replacing 20 with 4 and 5 with 1.

Similarly, 8 and 12 have a common factor of 4. Let me see. If I divide 8 by 4, it becomes 2, and divide 12 by 4, it becomes 3.

So let me do that step by step.

First, let me write the fractions again:

(1/5) * (8/7) * (20/12)

Let me rearrange the numerators and denominators to see common factors:

Numerators: 1, 8, 20

Denominators: 5, 7, 12

So, let me write them as:

(1 * 8 * 20) / (5 * 7 * 12)

Now, let me factor each number into primes:

1 is 1.

8 is 2^3.

20 is 2^2 * 5.

Denominator:

5 is 5.

7 is 7.

12 is 2^2 * 3.

So, combining all numerators:

Numerator factors: 1 * 2^3 * 2^2 * 5 = 2^(3+2) * 5 = 2^5 * 5

Denominator factors: 5 * 7 * 2^2 * 3 = 2^2 * 3 * 5 * 7

So, the entire fraction is (2^5 * 5) / (2^2 * 3 * 5 * 7)

Now, let's cancel common factors.

First, 2^5 divided by 2^2 is 2^(5-2) = 2^3.

Then, 5 in the numerator and 5 in the denominator cancel out.

So, remaining factors in numerator: 2^3

Remaining factors in denominator: 3 * 7

Therefore, the simplified fraction is 2^3 / (3 * 7) = 8 / 21.

So that's the same result as before. So that's reassuring.

Alternatively, maybe I can do step by step simplification. Let me try that.

Original expression: (1/5) * (8/7) * (20/12)

First, let me multiply (1/5) and (8/7).

Multiplying those two: (1*8)/(5*7) = 8/35.

Then, multiply that by (20/12). So, 8/35 * 20/12.

Again, let me simplify before multiplying.

Looking at 8 and 12: common factor of 4. 8 ÷ 4 = 2, 12 ÷ 4 = 3.

Looking at 20 and 35: common factor of 5. 20 ÷ 5 = 4, 35 ÷ 5 = 7.

So, simplifying:

(8/35) * (20/12) = (2/7) * (4/3) = (2*4)/(7*3) = 8/21.

Same result.

So, regardless of the method, I get 8/21.

Wait, let me check once more. Maybe I can compute decimal values to verify.

Original expression: (1/5) * (8/7) ÷ (12/20).

Compute each part:

1/5 is 0.2

8/7 is approximately 1.142857...

12/20 is 0.6, so dividing by 0.6 is the same as multiplying by 1/0.6 ≈ 1.666666...

So, let me compute step by step:

First, 0.2 * 1.142857 ≈ 0.2285714

Then, divide by 0.6: 0.2285714 / 0.6 ≈ 0.380952...

Now, 8/21 is approximately 0.380952... which matches.

So that's correct.

Alternatively, compute 8 divided by 21. 21*0.38 is 7.98, so yes, approximately 0.38095.

Therefore, the simplified fraction is 8/21.

So, I think that's the correct answer.

But just to recap all steps once again to make sure I didn't skip anything:

Original problem: (1/5) * (8/7) ÷ (12/20)

Convert division to multiplication by reciprocal: (1/5) * (8/7) * (20/12)

Multiply numerators: 1*8*20 = 160

Multiply denominators: 5*7*12 = 420

Simplify 160/420 by dividing numerator and denominator by 10: 16/42

Then divide by 2: 8/21.

Yes, that's correct.

Alternatively, simplifying before multiplying:

(1/5)*(8/7)*(20/12) = (1/5)*(8/7)*(5/3) [Wait, 20/12 simplifies to 5/3? Wait, 20 divided by 4 is 5, 12 divided by 4 is 3. So yes, 20/12 = 5/3.

So, substituting that in, the expression becomes (1/5) * (8/7) * (5/3).

Now, notice that 1/5 and 5/3 have a common factor of 5.

So, 1/5 * 5/3 = (1*5)/(5*3) = 5/15 = 1/3.

Then, multiply by 8/7: 1/3 * 8/7 = 8/21.

Oh, that's another way to simplify. So that's even quicker.

So, that's another method.

So, that's even better. So, if I simplify 20/12 first to 5/3, then the 5 in the denominator cancels with the 5 in the numerator from 1/5 * 5/3.

So, that's a more efficient way.

So, that's why it's important to simplify before multiplying.

So, in conclusion, regardless of the method, the answer is 8/21.

Therefore, the simplified form of the given expression is 8/21.

**Final Answer**
The simplified expression is \boxed{\dfrac{8}{21}}.

To simplify the expression:

$$
\frac{1}{5} \cdot \frac{8}{7} \div \frac{12}{20}
$$

---

### Step 1: Convert Division to Multiplication

Recall that dividing by a fraction is the same as multiplying by its reciprocal. So:

$$
\frac{1}{5} \cdot \frac{8}{7} \div \frac{12}{20} = \frac{1}{5} \cdot \frac{8}{7} \cdot \frac{20}{12}
$$

---

### Step 2: Multiply the Fractions

Now, multiply the numerators and denominators:

- Numerators: $1 \cdot 8 \cdot 20 = 160$
- Denominators: $5 \cdot 7 \cdot 12 = 420$

So the expression becomes:

$$
\frac{160}{420}
$$

---

### Step 3: Simplify the Fraction

To simplify $\frac{160}{420}$, find the greatest common divisor (GCD) of 160 and 420. The GCD is 20.

$$
\frac{160 \div 20}{420 \div 20} = \frac{8}{21}
$$

---

### Step 4: Final Answer

The simplified form of the expression is:

$$
\boxed{\dfrac{8}{21}}
$$

Solution: First, we can use the division rule so that we have an expression with only multiplication of fractions.  We get $$\frac{1}{5}\cdot \frac{8}{7}\div \frac{12}{20}=\frac{1}{5}\cdot \frac{8}{7}\cdot \frac{20}{12}.$$Now, notice that $5$ and $20$ have a common factor of $5$.  We can also see that $8$ and $12$ have a common factor of $4$.  Therefore, we can simplify to get $$\frac{1}{5}\cdot \frac{8}{7}\cdot \frac{20}{12}=\frac{1}{\cancel{5}}\cdot \frac{\cancelto{2}{8}}{7}\cdot \frac{\cancelto{4}{20}}{\cancelto{3}{12}}=\frac{1\cdot 2 \cdot 4}{7\cdot 3}=\boxed{\frac{8}{21}}.$$