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 affects the count because swapping boxes or balls wouldn't create a new arrangement.

Let me recall some combinatorial concepts. If the balls were distinguishable and the boxes were distinguishable, the number of ways would be 2^5, since each ball can go into either box. But here, both are indistinct, so that approach doesn't apply directly.

Wait, maybe I should think about partitions. Since the boxes are indistinct, the order of the boxes doesn't matter. So, for example, putting 3 balls in one box and 2 in the other is the same as putting 2 in one and 3 in the other. Therefore, I need to count the number of distinct partitions of 5 balls into up to 2 boxes.

But how do I count these partitions? Let me think. For indistinct boxes, the number of ways to distribute n indistinct objects into k indistinct boxes is equal to the number of integer partitions of n into at most k parts.

So here, n is 5 and k is 2. The integer partitions of 5 into at most 2 parts. Let me list them.

The partitions of 5 into 1 part: just [5].

Partitions into 2 parts: [4,1], [3,2].

Wait, is that all? Let me check.

Starting with the largest part. The largest part can be 5, which would mean the other part is 0. But since we are considering up to 2 parts, maybe 0 is allowed? Wait, but if the boxes are indistinct, can a box be empty?

The problem says "put 5 balls in 2 boxes". It doesn't specify whether boxes can be empty or not. Hmm. Wait, in some problems, boxes can be empty, but sometimes they can't. But since the problem doesn't specify, I need to assume that boxes can be empty.

But in the case of partitions, when we talk about partitions into at most k parts, we usually allow parts to be zero? Wait, no. Actually, integer partitions typically don't include zeros. So maybe I need to adjust for that.

Alternatively, maybe I should think of it as the number of ways to distribute the balls where the order of the boxes doesn't matter, and boxes can be empty.

So, for example, if we have two boxes, the possible distributions are:

- All 5 balls in one box, and 0 in the other. Since the boxes are indistinct, this is only one way.

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

- 3 balls in one box, 2 in the other. Another way.

Is there another? Let's see. If we try 2 and 3, that's the same as 3 and 2. Similarly, 1 and 4 is same as 4 and 1. So, the distinct distributions are:

1. (5,0)

2. (4,1)

3. (3,2)

So that's 3 ways. Therefore, the answer is 3?

Wait, but let me check again. Let me think of another approach.

Alternatively, if the boxes are indistinct, the number of ways is equal to the number of partitions of 5 into at most 2 parts. Which is exactly what I listed above.

But let me recall the formula for the number of ways to distribute n indistinct objects into k indistinct boxes.

I think it's equal to the number of partitions of n into at most k parts.

Alternatively, if boxes can be empty, then it's the number of partitions of n into at most k parts.

But if boxes cannot be empty, it's the number of partitions of n into exactly k parts.

But in our case, since the problem says "put 5 balls in 2 boxes", it's possible that boxes can be empty. So, the answer would be the number of partitions of 5 into at most 2 parts.

Which is 3, as above.

But let me check with another method.

Suppose we think of the problem as equivalent to finding the number of equivalence classes under the action of swapping boxes.

If the boxes were distinguishable, there would be C(5 + 2 -1, 2 -1) = C(6,1) = 6 ways. Wait, that's the stars and bars formula. But since the boxes are indistinct, we need to adjust for overcounting.

But how does that work?

In stars and bars, when boxes are distinguishable, the number is C(n + k -1, k -1). For n=5, k=2, that's C(5 + 2 -1, 2 -1) = C(6,1) = 6.

But since the boxes are indistinct, each distribution is counted twice in the stars and bars count, except when the distribution is symmetric (i.e., when the number of balls in each box is the same).

But in our case, since 5 is odd, there is no symmetric distribution. Because 5 divided by 2 is 2.5, which is not an integer. Therefore, all distributions are counted twice.

Therefore, the number of distinct distributions would be 6 divided by 2, which is 3.

So that gives the same answer as before.

Therefore, the answer is 3.

But let me check again with another approach.

Suppose I list all possible distributions.

Since the boxes are indistinct, we can assume that the number of balls in the first box is less than or equal to the number in the second.

So, the possible distributions are:

- 0 and 5

- 1 and 4

- 2 and 3

These are the only possibilities where the first number is less than or equal to the second.

So that's 3 distributions.

Therefore, the answer is 3.

But let me think again if there's a possibility that I'm missing something.

Wait, if the boxes are indistinct, then putting all balls in one box is considered one way, regardless of which box it is. Similarly, putting 1 and 4 is one way.

Alternatively, if the boxes were distinct, then for each distribution, there would be two possibilities unless the numbers are equal.

But since they are indistinct, we have to consider that.

So, for example, if we have 5 balls and 2 boxes, the number of ways is equal to the number of integer solutions to a + b = 5, where a and b are non-negative integers, and considering that (a,b) is the same as (b,a).

So, how many such unordered pairs are there?

Let me think.

The number of unordered pairs (a,b) with a + b = 5, where a ≤ b.

So, starting with a=0, then b=5.

a=1, b=4.

a=2, b=3.

a=3, b=2. But since a ≤ b, we stop at a=2.

So, there are 3 such pairs.

Therefore, the answer is 3.

Therefore, all methods lead to 3.

But let me check with another formula.

I remember that the number of ways to put n indistinct objects into k indistinct boxes is equal to the partition function P(n, k), which is the number of partitions of n into at most k parts.

Alternatively, it's also equal to the number of partitions of n into exactly k non-negative integers, where order doesn't matter.

But in our case, since boxes can be empty, it's the number of partitions of 5 into at most 2 parts.

The partition function P(5, 2) is indeed 3.

Let me check the partition function.

The number of partitions of 5 is:

- 5

- 4 + 1

- 3 + 2

- 3 + 1 + 1

- 2 + 2 + 1

- 2 + 1 + 1 + 1

- 1 + 1 + 1 + 1 + 1

But P(5, 2) is the number of partitions of 5 into at most 2 parts. So, that would be the first three: 5, 4+1, 3+2.

Yes, so that's 3.

Therefore, the answer is 3.

Alternatively, if I think of the formula for the number of ways to distribute n indistinct objects into k indistinct boxes, it's equal to the floor(n/2) + 1? Wait, no. Wait, for n=5 and k=2, floor(5/2) is 2, plus 1 is 3. So that works here.

But is that a general formula? For example, if n=4 and k=2, then floor(4/2) +1 = 2 +1=3. But the partitions of 4 into at most 2 parts are 4, 3+1, 2+2. So that's 3, which matches.

Similarly, for n=3, floor(3/2) +1=1+1=2. The partitions are 3, 2+1. Which is 2.

So, seems like for k=2, the number of ways is floor(n/2) +1? Wait, but for n=5, floor(5/2)=2, so 2+1=3. For n=6, floor(6/2)=3, so 3+1=4. Let's check: partitions of 6 into at most 2 parts: 6, 5+1, 4+2, 3+3. That's 4. So yes, that works.

So, in general, for k=2, the number of ways is floor(n/2) +1? Wait, but actually, if you think about it, the number of partitions of n into at most 2 parts is equal to the number of integers m such that 0 ≤ m ≤ floor(n/2). Because for each m from 0 to floor(n/2), you can have a partition of n as (n - m, m).

Wait, but when m=0, it's (n, 0). When m=1, (n-1, 1). Up to m= floor(n/2).

So, the number of such partitions is floor(n/2) +1.

Wait, for n=5, floor(5/2)=2, so 2+1=3. Which matches.

So, in general, for two boxes, the number of ways is floor(n/2) +1.

But wait, but in our case, the boxes are indistinct, so we can have (n, 0), (n-1, 1), ..., up to (floor(n/2) + ceil(n/2), ...).

But since the boxes are indistinct, we don't care about the order.

Therefore, the formula floor(n/2) +1 gives the number of ways.

But let me check for n=2. floor(2/2) +1=1+1=2. The partitions are (2,0) and (1,1). Which is 2. Correct.

For n=1, floor(1/2)=0, so 0+1=1. Which is (1,0). Correct.

So, yes, this formula works.

Therefore, for n=5, floor(5/2)=2, so 2+1=3.

Therefore, the answer is 3.

But let me think again if there's another way someone might approach this problem and get confused.

For example, someone might think that since the boxes are indistinct, the number of ways is equal to the number of ways to choose how many balls go into one box, divided by something.

But since the boxes are indistinct, choosing to put 0 in one and 5 in the other is the same as putting 5 in one and 0 in the other. So, if you think of it as choosing a number of balls for one box, you have to avoid double-counting.

So, if you consider the number of balls in one box, it can be from 0 to 5. But since the boxes are indistinct, each distribution is counted twice except when the number of balls is the same in both boxes.

But since 5 is odd, there is no such case. Therefore, the total number of distributions is (number of distributions with boxes distinguishable)/2.

Which is 6/2=3.

So that's another way to think about it.

Alternatively, if the boxes were distinguishable, the number of ways is C(5 + 2 -1, 2 -1)=C(6,1)=6.

But since they are indistinct, divide by 2, getting 3.

But why divide by 2? Because each distribution is counted twice in the distinguishable case.

But if there was a case where the distribution is symmetric (i.e., same number of balls in both boxes), then that case would be counted once, and the others would be counted twice.

But since 5 is odd, there is no symmetric distribution.

Therefore, all distributions are counted twice, so dividing by 2 gives the correct count.

Therefore, 6/2=3.

So, again, same answer.

Therefore, I think the answer is 3.

But let me check with a concrete enumeration.

Suppose the boxes are Box A and Box B, but they are indistinct.

So, the possible distributions are:

1. All balls in Box A, none in Box B. But since boxes are indistinct, this is same as all balls in Box B, none in Box A. So, only one way.

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

3. 3 balls in one box, 2 in the other. Another way.

Is there another way? Let's see.

If we try 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, no other possibilities.

Therefore, only 3 ways.

Therefore, I think the answer is 3.

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

Suppose I use generating functions.

The generating function for the number of ways to put indistinct balls into indistinct boxes is the product for each box of (1 + x + x^2 + ...). But since the boxes are indistinct, we need to consider the generating function for partitions.

Wait, actually, generating functions for partitions into at most k parts.

The generating function for partitions into at most 2 parts is:

(1 + x + x^2 + x^3 + ...)(1 + x^2 + x^3 + x^4 + ...) ?

Wait, no. Wait, the generating function for partitions into at most 2 parts is the same as the generating function for partitions where each part is at most 2? No, that's different.

Wait, actually, the generating function for partitions into at most 2 parts is:

Sum_{n=0}^\infty p(n, 2) x^n, where p(n, 2) is the number of partitions of n into at most 2 parts.

This generating function is:

(1 + x + x^2 + x^3 + ...)(1 + x^2 + x^3 + x^4 + ...) ?

No, actually, the generating function for partitions into at most 2 parts is:

(1 + x + x^2 + x^3 + ...)(1 + x + x^2 + x^3 + ...) ?

Wait, no. Wait, if we think of partitions into at most 2 parts, then each partition is either a single part (so 1 + x + x^2 + ...) or two parts where the first part is at least the second.

But generating functions can be a bit tricky here.

Alternatively, since the number of partitions of n into at most 2 parts is equal to the floor(n/2) + 1, as we saw earlier, then the generating function would be:

Sum_{n=0}^\infty (floor(n/2) + 1) x^n.

But maybe that's not helpful here.

Alternatively, since we know that for n=5, the number is 3, and we have already confirmed that through multiple methods, maybe this is overcomplicating.

Alternatively, think of the problem as equivalent to the number of equivalence classes under the action of swapping boxes.

In the case of distinguishable boxes, there are 6 distributions. Each equivalence class has size 2, except if the distribution is symmetric. Since there are no symmetric distributions here, all 6 are in pairs. Therefore, 6 / 2 = 3.

Therefore, again, same answer.

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

But just to make sure, let me check with a small example.

Suppose n=1, k=2. How many ways?

Since the balls are indistinct and boxes are indistinct. The only way is to put the ball in one box and leave the other empty. So, 1 way.

Using the formula floor(1/2) +1 = 0 +1=1. Correct.

Another example: n=2, k=2.

Possible distributions: (2,0) and (1,1). So 2 ways.

Formula: floor(2/2) +1=1 +1=2. Correct.

Another example: n=3, k=2.

Distributions: (3,0), (2,1). So 2 ways.

Formula: floor(3/2) +1=1 +1=2. Correct.

n=4, k=2: (4,0), (3,1), (2,2). So 3 ways.

Formula: floor(4/2) +1=2 +1=3. Correct.

n=5, k=2: 3 ways.

So, yes, the formula works.

Therefore, the answer is 3.

Therefore, I think the answer is \boxed{3}.

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

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

---

### 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**: Swapping the boxes does not result in a new configuration. So, the arrangement (3 balls in one box, 2 in the other) is the same as (2 in one, 3 in the other).

---

### 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 the boxes are indistinguishable, and we are only interested in the **distribution of balls**, not which box has how many.

---

### Step-by-Step Enumeration:

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

1. **(5, 0)**: All 5 balls in one box, 0 in the other.
2. **(4, 1)**: 4 balls in one box, 1 in the other.
3. **(3, 2)**: 3 balls in one box, 2 in the other.

These are all the distinct ways to split 5 into two non-negative integers, where the order does not matter. There are **no more** such partitions, as any other combination would either repeat one of these or violate the "at most 2 parts" condition.

---

### Alternative Approach: Using the Partition Function

The number of ways to distribute $ n $ indistinguishable objects into $ k $ indistinguishable boxes is equal to the number of **integer partitions of $ n $ into at most $ k $ parts**.

In this case, $ n = 5 $, $ k = 2 $, and the number of such partitions is:

$$
P(5, 2) = 3
$$

This confirms our earlier enumeration.

---

### 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 affects the count because swapping boxes or balls wouldn't create a new arrangement.

Let me recall some combinatorial concepts. If the balls were distinguishable and the boxes were distinguishable, the number of ways would be 2^5, since each ball can go into either box. But here, both are indistinct, so that approach doesn't apply directly.

Wait, maybe I should think about partitions. Since the boxes are indistinct, the order of the boxes doesn't matter. So, for example, putting 3 balls in one box and 2 in the other is the same as putting 2 in one and 3 in the other. Therefore, I need to count the number of distinct partitions of 5 balls into up to 2 boxes.

But how do I count these partitions? Let me think. For indistinct boxes, the number of ways to distribute n indistinct objects into k indistinct boxes is equal to the number of integer partitions of n into at most k parts.

So here, n is 5 and k is 2. The integer partitions of 5 into at most 2 parts. Let me list them.

The partitions of 5 into 1 part: just [5].

Partitions into 2 parts: [4,1], [3,2].

Wait, is that all? Let me check.

Starting with the largest part. The largest part can be 5, which would mean the other part is 0. But since we are considering up to 2 parts, maybe 0 is allowed? Wait, but if the boxes are indistinct, can a box be empty?

The problem says "put 5 balls in 2 boxes". It doesn't specify whether boxes can be empty or not. Hmm. Wait, in some problems, boxes can be empty, but sometimes they can't. But since the problem doesn't specify, I need to assume that boxes can be empty.

But in the case of partitions, when we talk about partitions into at most k parts, we usually allow parts to be zero? Wait, no. Actually, integer partitions typically don't include zeros. So maybe I need to adjust for that.

Alternatively, maybe I should think of it as the number of ways to distribute the balls where the order of the boxes doesn't matter, and boxes can be empty.

So, for example, if we have two boxes, the possible distributions are:

- All 5 balls in one box, and 0 in the other. Since the boxes are indistinct, this is only one way.

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

- 3 balls in one box, 2 in the other. Another way.

Is there another? Let's see. If we try 2 and 3, that's the same as 3 and 2. Similarly, 1 and 4 is same as 4 and 1. So, the distinct distributions are:

1. (5,0)

2. (4,1)

3. (3,2)

So that's 3 ways. Therefore, the answer is 3?

Wait, but let me check again. Let me think of another approach.

Alternatively, if the boxes are indistinct, the number of ways is equal to the number of partitions of 5 into at most 2 parts. Which is exactly what I listed above.

But let me recall the formula for the number of ways to distribute n indistinct objects into k indistinct boxes.

I think it's equal to the number of partitions of n into at most k parts.

Alternatively, if boxes can be empty, then it's the number of partitions of n into at most k parts.

But if boxes cannot be empty, it's the number of partitions of n into exactly k parts.

But in our case, since the problem says "put 5 balls in 2 boxes", it's possible that boxes can be empty. So, the answer would be the number of partitions of 5 into at most 2 parts.

Which is 3, as above.

But let me check with another method.

Suppose we think of the problem as equivalent to finding the number of equivalence classes under the action of swapping boxes.

If the boxes were distinguishable, there would be C(5 + 2 -1, 2 -1) = C(6,1) = 6 ways. Wait, that's the stars and bars formula. But since the boxes are indistinct, we need to adjust for overcounting.

But how does that work?

In stars and bars, when boxes are distinguishable, the number is C(n + k -1, k -1). For n=5, k=2, that's C(5 + 2 -1, 2 -1) = C(6,1) = 6.

But since the boxes are indistinct, each distribution is counted twice in the stars and bars count, except when the distribution is symmetric (i.e., when the number of balls in each box is the same).

But in our case, since 5 is odd, there is no symmetric distribution. Because 5 divided by 2 is 2.5, which is not an integer. Therefore, all distributions are counted twice.

Therefore, the number of distinct distributions would be 6 divided by 2, which is 3.

So that gives the same answer as before.

Therefore, the answer is 3.

But let me check again with another approach.

Suppose I list all possible distributions.

Since the boxes are indistinct, we can assume that the number of balls in the first box is less than or equal to the number in the second.

So, the possible distributions are:

- 0 and 5

- 1 and 4

- 2 and 3

These are the only possibilities where the first number is less than or equal to the second.

So that's 3 distributions.

Therefore, the answer is 3.

But let me think again if there's a possibility that I'm missing something.

Wait, if the boxes are indistinct, then putting all balls in one box is considered one way, regardless of which box it is. Similarly, putting 1 and 4 is one way.

Alternatively, if the boxes were distinct, then for each distribution, there would be two possibilities unless the numbers are equal.

But since they are indistinct, we have to consider that.

So, for example, if we have 5 balls and 2 boxes, the number of ways is equal to the number of integer solutions to a + b = 5, where a and b are non-negative integers, and considering that (a,b) is the same as (b,a).

So, how many such unordered pairs are there?

Let me think.

The number of unordered pairs (a,b) with a + b = 5, where a ≤ b.

So, starting with a=0, then b=5.

a=1, b=4.

a=2, b=3.

a=3, b=2. But since a ≤ b, we stop at a=2.

So, there are 3 such pairs.

Therefore, the answer is 3.

Therefore, all methods lead to 3.

But let me check with another formula.

I remember that the number of ways to put n indistinct objects into k indistinct boxes is equal to the partition function P(n, k), which is the number of partitions of n into at most k parts.

Alternatively, it's also equal to the number of partitions of n into exactly k non-negative integers, where order doesn't matter.

But in our case, since boxes can be empty, it's the number of partitions of 5 into at most 2 parts.

The partition function P(5, 2) is indeed 3.

Let me check the partition function.

The number of partitions of 5 is:

- 5

- 4 + 1

- 3 + 2

- 3 + 1 + 1

- 2 + 2 + 1

- 2 + 1 + 1 + 1

- 1 + 1 + 1 + 1 + 1

But P(5, 2) is the number of partitions of 5 into at most 2 parts. So, that would be the first three: 5, 4+1, 3+2.

Yes, so that's 3.

Therefore, the answer is 3.

Alternatively, if I think of the formula for the number of ways to distribute n indistinct objects into k indistinct boxes, it's equal to the floor(n/2) + 1? Wait, no. Wait, for n=5 and k=2, floor(5/2) is 2, plus 1 is 3. So that works here.

But is that a general formula? For example, if n=4 and k=2, then floor(4/2) +1 = 2 +1=3. But the partitions of 4 into at most 2 parts are 4, 3+1, 2+2. So that's 3, which matches.

Similarly, for n=3, floor(3/2) +1=1+1=2. The partitions are 3, 2+1. Which is 2.

So, seems like for k=2, the number of ways is floor(n/2) +1? Wait, but for n=5, floor(5/2)=2, so 2+1=3. For n=6, floor(6/2)=3, so 3+1=4. Let's check: partitions of 6 into at most 2 parts: 6, 5+1, 4+2, 3+3. That's 4. So yes, that works.

So, in general, for k=2, the number of ways is floor(n/2) +1? Wait, but actually, if you think about it, the number of partitions of n into at most 2 parts is equal to the number of integers m such that 0 ≤ m ≤ floor(n/2). Because for each m from 0 to floor(n/2), you can have a partition of n as (n - m, m).

Wait, but when m=0, it's (n, 0). When m=1, (n-1, 1). Up to m= floor(n/2).

So, the number of such partitions is floor(n/2) +1.

Wait, for n=5, floor(5/2)=2, so 2+1=3. Which matches.

So, in general, for two boxes, the number of ways is floor(n/2) +1.

But wait, but in our case, the boxes are indistinct, so we can have (n, 0), (n-1, 1), ..., up to (floor(n/2) + ceil(n/2), ...).

But since the boxes are indistinct, we don't care about the order.

Therefore, the formula floor(n/2) +1 gives the number of ways.

But let me check for n=2. floor(2/2) +1=1+1=2. The partitions are (2,0) and (1,1). Which is 2. Correct.

For n=1, floor(1/2)=0, so 0+1=1. Which is (1,0). Correct.

So, yes, this formula works.

Therefore, for n=5, floor(5/2)=2, so 2+1=3.

Therefore, the answer is 3.

But let me think again if there's another way someone might approach this problem and get confused.

For example, someone might think that since the boxes are indistinct, the number of ways is equal to the number of ways to choose how many balls go into one box, divided by something.

But since the boxes are indistinct, choosing to put 0 in one and 5 in the other is the same as putting 5 in one and 0 in the other. So, if you think of it as choosing a number of balls for one box, you have to avoid double-counting.

So, if you consider the number of balls in one box, it can be from 0 to 5. But since the boxes are indistinct, each distribution is counted twice except when the number of balls is the same in both boxes.

But since 5 is odd, there is no such case. Therefore, the total number of distributions is (number of distributions with boxes distinguishable)/2.

Which is 6/2=3.

So that's another way to think about it.

Alternatively, if the boxes were distinguishable, the number of ways is C(5 + 2 -1, 2 -1)=C(6,1)=6.

But since they are indistinct, divide by 2, getting 3.

But why divide by 2? Because each distribution is counted twice in the distinguishable case.

But if there was a case where the distribution is symmetric (i.e., same number of balls in both boxes), then that case would be counted once, and the others would be counted twice.

But since 5 is odd, there is no symmetric distribution.

Therefore, all distributions are counted twice, so dividing by 2 gives the correct count.

Therefore, 6/2=3.

So, again, same answer.

Therefore, I think the answer is 3.

But let me check with a concrete enumeration.

Suppose the boxes are Box A and Box B, but they are indistinct.

So, the possible distributions are:

1. All balls in Box A, none in Box B. But since boxes are indistinct, this is same as all balls in Box B, none in Box A. So, only one way.

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

3. 3 balls in one box, 2 in the other. Another way.

Is there another way? Let's see.

If we try 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, no other possibilities.

Therefore, only 3 ways.

Therefore, I think the answer is 3.

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

Suppose I use generating functions.

The generating function for the number of ways to put indistinct balls into indistinct boxes is the product for each box of (1 + x + x^2 + ...). But since the boxes are indistinct, we need to consider the generating function for partitions.

Wait, actually, generating functions for partitions into at most k parts.

The generating function for partitions into at most 2 parts is:

(1 + x + x^2 + x^3 + ...)(1 + x^2 + x^3 + x^4 + ...) ?

Wait, no. Wait, the generating function for partitions into at most 2 parts is the same as the generating function for partitions where each part is at most 2? No, that's different.

Wait, actually, the generating function for partitions into at most 2 parts is:

Sum_{n=0}^\infty p(n, 2) x^n, where p(n, 2) is the number of partitions of n into at most 2 parts.

This generating function is:

(1 + x + x^2 + x^3 + ...)(1 + x^2 + x^3 + x^4 + ...) ?

No, actually, the generating function for partitions into at most 2 parts is:

(1 + x + x^2 + x^3 + ...)(1 + x + x^2 + x^3 + ...) ?

Wait, no. Wait, if we think of partitions into at most 2 parts, then each partition is either a single part (so 1 + x + x^2 + ...) or two parts where the first part is at least the second.

But generating functions can be a bit tricky here.

Alternatively, since the number of partitions of n into at most 2 parts is equal to the floor(n/2) + 1, as we saw earlier, then the generating function would be:

Sum_{n=0}^\infty (floor(n/2) + 1) x^n.

But maybe that's not helpful here.

Alternatively, since we know that for n=5, the number is 3, and we have already confirmed that through multiple methods, maybe this is overcomplicating.

Alternatively, think of the problem as equivalent to the number of equivalence classes under the action of swapping boxes.

In the case of distinguishable boxes, there are 6 distributions. Each equivalence class has size 2, except if the distribution is symmetric. Since there are no symmetric distributions here, all 6 are in pairs. Therefore, 6 / 2 = 3.

Therefore, again, same answer.

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

But just to make sure, let me check with a small example.

Suppose n=1, k=2. How many ways?

Since the balls are indistinct and boxes are indistinct. The only way is to put the ball in one box and leave the other empty. So, 1 way.

Using the formula floor(1/2) +1 = 0 +1=1. Correct.

Another example: n=2, k=2.

Possible distributions: (2,0) and (1,1). So 2 ways.

Formula: floor(2/2) +1=1 +1=2. Correct.

Another example: n=3, k=2.

Distributions: (3,0), (2,1). So 2 ways.

Formula: floor(3/2) +1=1 +1=2. Correct.

n=4, k=2: (4,0), (3,1), (2,2). So 3 ways.

Formula: floor(4/2) +1=2 +1=3. Correct.

n=5, k=2: 3 ways.

So, yes, the formula works.

Therefore, the answer is 3.

Therefore, I think the answer is \boxed{3}.

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

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

---

### 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**: Swapping the boxes does not result in a new configuration. So, the arrangement (3 balls in one box, 2 in the other) is the same as (2 in one, 3 in the other).

---

### 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 the boxes are indistinguishable, and we are only interested in the **distribution of balls**, not which box has how many.

---

### Step-by-Step Enumeration:

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

1. **(5, 0)**: All 5 balls in one box, 0 in the other.
2. **(4, 1)**: 4 balls in one box, 1 in the other.
3. **(3, 2)**: 3 balls in one box, 2 in the other.

These are all the distinct ways to split 5 into two non-negative integers, where the order does not matter. There are **no more** such partitions, as any other combination would either repeat one of these or violate the "at most 2 parts" condition.

---

### Alternative Approach: Using the Partition Function

The number of ways to distribute $ n $ indistinguishable objects into $ k $ indistinguishable boxes is equal to the number of **integer partitions of $ n $ into at most $ k $ parts**.

In this case, $ n = 5 $, $ k = 2 $, and the number of such partitions is:

$$
P(5, 2) = 3
$$

This confirms our earlier enumeration.

---

### 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. If it was something like r = z, that would be a cone. But here, z is fixed.

Alternatively, maybe someone could confuse it with a sphere? But a sphere would have an equation like r = something involving z. For example, in cylindrical coordinates, a sphere of radius a centered at the origin would be r² + z² = a². So, that's different.

Alternatively, if you have z = c, then in Cartesian coordinates, that's z = c, which is a plane. So, unless there's some other interpretation, I think it's definitely a plane.

Wait, but maybe in cylindrical coordinates, z = c is a horizontal line? But no, in 3D space, a line is one-dimensional. But z = c is a two-dimensional surface, because you can vary r and θ freely. So, it's a plane.

Alternatively, if you fix z = c and θ = something, then it would be a line. But here, only z is fixed. So, the answer is (C) Plane.

Let me check the options again. The options are:

(A) Line

(B) Circle

(C) Plane

(D) Sphere

(E) Cylinder

(F) Cone

So, (C) Plane is the answer.

But just to make sure, let me think of specific examples. For instance, if c = 0, then z = 0 is the xy-plane, which is a plane. If c = 5, then it's a plane parallel to the xy-plane at height 5. So, yes, that's definitely a plane.

Alternatively, if someone thought that in cylindrical coordinates, z = c might be a cylinder, but no. A cylinder would be something like r = c, which is a cylinder extending along the z-axis. But z = c is a plane.

Therefore, I think the correct 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 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,
- The surface extends infinitely in the radial and angular directions (i.e., for all $r \geq 0$ and all $\theta$).

This is the definition of a **plane** in 3D space. Specifically, it is a **horizontal plane** that is parallel to the $xy$-plane and intersects the $z$-axis at $z = c$.

---

### Comparison with Other Options

Let’s briefly consider the other options to ensure that $z = c$ is not misinterpreted:

- **(A) Line**: A line is one-dimensional. The equation $z = c$ defines a two-dimensional surface, so this is not a line.
- **(B) Circle**: A circle is a closed curve in 2D. The equation $z = c$ is not a curve but a surface.
- **(C) Plane**: As discussed, this is the correct interpretation.
- **(D) Sphere**: A sphere would have an equation like $r^2 + z^2 = a^2$, which is not the case here.
- **(E) Cylinder**: A cylinder is defined by a fixed $r$, such as $r = c$, not a fixed $z$.
- **(F) Cone**: A cone would involve a relationship between $r$ and $z$, such as $r = kz$, which is not the case here.

---

### Final Conclusion

The equation $z = c$ in cylindrical coordinates represents a **plane** that is parallel to the $xy$-plane and located at height $z = c$. This is consistent with the same equation in Cartesian coordinates.

$$
\boxed{C}
$$

[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. If it was something like r = z, that would be a cone. But here, z is fixed.

Alternatively, maybe someone could confuse it with a sphere? But a sphere would have an equation like r = something involving z. For example, in cylindrical coordinates, a sphere of radius a centered at the origin would be r² + z² = a². So, that's different.

Alternatively, if you have z = c, then in Cartesian coordinates, that's z = c, which is a plane. So, unless there's some other interpretation, I think it's definitely a plane.

Wait, but maybe in cylindrical coordinates, z = c is a horizontal line? But no, in 3D space, a line is one-dimensional. But z = c is a two-dimensional surface, because you can vary r and θ freely. So, it's a plane.

Alternatively, if you fix z = c and θ = something, then it would be a line. But here, only z is fixed. So, the answer is (C) Plane.

Let me check the options again. The options are:

(A) Line

(B) Circle

(C) Plane

(D) Sphere

(E) Cylinder

(F) Cone

So, (C) Plane is the answer.

But just to make sure, let me think of specific examples. For instance, if c = 0, then z = 0 is the xy-plane, which is a plane. If c = 5, then it's a plane parallel to the xy-plane at height 5. So, yes, that's definitely a plane.

Alternatively, if someone thought that in cylindrical coordinates, z = c might be a cylinder, but no. A cylinder would be something like r = c, which is a cylinder extending along the z-axis. But z = c is a plane.

Therefore, I think the correct 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 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,
- The surface extends infinitely in the radial and angular directions (i.e., for all $r \geq 0$ and all $\theta$).

This is the definition of a **plane** in 3D space. Specifically, it is a **horizontal plane** that is parallel to the $xy$-plane and intersects the $z$-axis at $z = c$.

---

### Comparison with Other Options

Let’s briefly consider the other options to ensure that $z = c$ is not misinterpreted:

- **(A) Line**: A line is one-dimensional. The equation $z = c$ defines a two-dimensional surface, so this is not a line.
- **(B) Circle**: A circle is a closed curve in 2D. The equation $z = c$ is not a curve but a surface.
- **(C) Plane**: As discussed, this is the correct interpretation.
- **(D) Sphere**: A sphere would have an equation like $r^2 + z^2 = a^2$, which is not the case here.
- **(E) Cylinder**: A cylinder is defined by a fixed $r$, such as $r = c$, not a fixed $z$.
- **(F) Cone**: A cone would involve a relationship between $r$ and $z$, such as $r = kz$, which is not the case here.

---

### Final Conclusion

The equation $z = c$ in cylindrical coordinates represents a **plane** that is parallel to the $xy$-plane and located at height $z = c$. This is consistent with the same equation in Cartesian coordinates.

$$
\boxed{C}
$$

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 (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, let me write the factors:

(x - 4 + sqrt(11))(x - 4 - sqrt(11)). Wait, that's actually (x - 4)^2 - (sqrt(11))^2, right? Because it's of the form (a + b)(a - b) = a² - b². So here, a is (x - 4) and b is sqrt(11). Therefore, expanding that gives:

(x - 4)^2 - (sqrt(11))^2 = (x² - 8x + 16) - 11 = x² - 8x + 16 - 11 = x² - 8x + 5.

So, p(x) is x² - 8x + 5. Wait, but the problem says that p(x) has integer coefficients, which this does. So, is this the polynomial? But maybe there's a scalar multiple? Because if there's a leading coefficient, say, k, then p(x) could be k(x² - 8x + 5). However, since the problem says "quadratic polynomial with integer coefficients," and doesn't specify that it's monic, but since they want p(3)/p(4), maybe the scalar multiple cancels out? Let me check.

Suppose p(x) = k(x² - 8x + 5). Then p(3) = k(9 - 24 + 5) = k(-10), and p(4) = k(16 - 32 + 5) = k(-11). Therefore, p(3)/p(4) would be (-10k)/(-11k) = 10/11. So regardless of the value of k, as long as k ≠ 0, which it can't be because it's a quadratic polynomial, the ratio is 10/11. Therefore, the answer is 10/11?

Wait, but let me make sure that there isn't another possible polynomial. For example, could there be another quadratic polynomial with integer coefficients that has 4 - sqrt(11) as a root but isn't just a scalar multiple of x² - 8x + 5? Well, since the minimal polynomial for 4 - sqrt(11) over the integers is x² - 8x + 5, because that's the monic polynomial with integer coefficients that has that root, and any other polynomial with integer coefficients that has that root must be a multiple of this minimal polynomial. Therefore, p(x) must be k(x² - 8x + 5) for some integer k. Therefore, the ratio p(3)/p(4) is indeed 10/11 regardless of k. Therefore, the answer is 10/11.

But let me verify this by actually computing p(3) and p(4) with the polynomial x² - 8x + 5. Let's compute p(3):

p(3) = 3² - 8*3 + 5 = 9 - 24 + 5 = (9 + 5) - 24 = 14 - 24 = -10.

p(4) = 4² - 8*4 + 5 = 16 - 32 + 5 = (16 + 5) - 32 = 21 - 32 = -11.

Therefore, p(3)/p(4) = (-10)/(-11) = 10/11. Yep, that checks out. So even if there was a scalar multiple, the ratio remains the same. Therefore, the answer is 10/11.

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 is x² - (sum)x + product = x² - 8x + 5. So same result. Therefore, same conclusion.

Alternatively, if someone didn't remember the conjugate root theorem, they might try to construct the polynomial. Let me try that approach. Suppose that 4 - sqrt(11) is a root. Then p(4 - sqrt(11)) = 0. Let me assume p(x) is ax² + bx + c, with a, b, c integers. Then:

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

Let me compute (4 - sqrt(11))²:

(4 - sqrt(11))² = 16 - 8 sqrt(11) + 11 = 27 - 8 sqrt(11).

Therefore, substituting back:

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

Expanding:

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:

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 ax² + bx + c = a x² - 8a x + 5a = a(x² - 8x + 5). So again, same result. Therefore, p(x) is a multiple of x² - 8x + 5, so the ratio p(3)/p(4) is 10/11. Therefore, regardless of the value of a, the ratio is the same. Therefore, the answer is 10/11.

Alternatively, maybe someone could think that the polynomial could have other roots? But since it's quadratic, it can only have two roots. Since we already have two roots, 4 ± sqrt(11), there are no other roots. Therefore, the polynomial is uniquely determined up to a scalar multiple. Therefore, the ratio is fixed.

Therefore, I think the answer is 10/11. Let me check once more with actual numbers. Let me take p(x) = x² - 8x + 5. Then p(3) = 9 - 24 + 5 = -10, p(4) = 16 - 32 + 5 = -11. So -10/-11 = 10/11. If I take p(x) = 2x² - 16x + 10, then p(3) = 2*9 - 16*3 + 10 = 18 - 48 + 10 = -20, p(4) = 2*16 - 16*4 + 10 = 32 - 64 + 10 = -22. Then -20/-22 = 10/11. Same result. So regardless of the scalar multiple, the ratio is 10/11. Therefore, the answer is 10/11.

I think that's solid. I don't see any mistakes in my reasoning. So I think 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 polynomial $ p(x) $ must have the form:

$$
p(x) = a(x - (4 - \sqrt{11}))(x - (4 + \sqrt{11}))
$$

for some integer $ a \neq 0 $.

---

### Step 2: Simplify the Polynomial

We simplify the expression:

$$
p(x) = a(x - 4 + \sqrt{11})(x - 4 - \sqrt{11}) = a[(x - 4)^2 - (\sqrt{11})^2]
$$

$$
= a[(x - 4)^2 - 11] = a(x^2 - 8x + 16 - 11) = a(x^2 - 8x + 5)
$$

So, the polynomial is:

$$
p(x) = a(x^2 - 8x + 5)
$$

---

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

We now compute the values of $ p(3) $ and $ p(4) $:

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

---

### Step 4: Compute the Ratio

Now, compute the ratio:

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

Note that the scalar $ a $ cancels out, as long as $ a \neq 0 $, which is required for $ p(x) $ to be a quadratic polynomial.

---

### 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 (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, let me write the factors:

(x - 4 + sqrt(11))(x - 4 - sqrt(11)). Wait, that's actually (x - 4)^2 - (sqrt(11))^2, right? Because it's of the form (a + b)(a - b) = a² - b². So here, a is (x - 4) and b is sqrt(11). Therefore, expanding that gives:

(x - 4)^2 - (sqrt(11))^2 = (x² - 8x + 16) - 11 = x² - 8x + 16 - 11 = x² - 8x + 5.

So, p(x) is x² - 8x + 5. Wait, but the problem says that p(x) has integer coefficients, which this does. So, is this the polynomial? But maybe there's a scalar multiple? Because if there's a leading coefficient, say, k, then p(x) could be k(x² - 8x + 5). However, since the problem says "quadratic polynomial with integer coefficients," and doesn't specify that it's monic, but since they want p(3)/p(4), maybe the scalar multiple cancels out? Let me check.

Suppose p(x) = k(x² - 8x + 5). Then p(3) = k(9 - 24 + 5) = k(-10), and p(4) = k(16 - 32 + 5) = k(-11). Therefore, p(3)/p(4) would be (-10k)/(-11k) = 10/11. So regardless of the value of k, as long as k ≠ 0, which it can't be because it's a quadratic polynomial, the ratio is 10/11. Therefore, the answer is 10/11?

Wait, but let me make sure that there isn't another possible polynomial. For example, could there be another quadratic polynomial with integer coefficients that has 4 - sqrt(11) as a root but isn't just a scalar multiple of x² - 8x + 5? Well, since the minimal polynomial for 4 - sqrt(11) over the integers is x² - 8x + 5, because that's the monic polynomial with integer coefficients that has that root, and any other polynomial with integer coefficients that has that root must be a multiple of this minimal polynomial. Therefore, p(x) must be k(x² - 8x + 5) for some integer k. Therefore, the ratio p(3)/p(4) is indeed 10/11 regardless of k. Therefore, the answer is 10/11.

But let me verify this by actually computing p(3) and p(4) with the polynomial x² - 8x + 5. Let's compute p(3):

p(3) = 3² - 8*3 + 5 = 9 - 24 + 5 = (9 + 5) - 24 = 14 - 24 = -10.

p(4) = 4² - 8*4 + 5 = 16 - 32 + 5 = (16 + 5) - 32 = 21 - 32 = -11.

Therefore, p(3)/p(4) = (-10)/(-11) = 10/11. Yep, that checks out. So even if there was a scalar multiple, the ratio remains the same. Therefore, the answer is 10/11.

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 is x² - (sum)x + product = x² - 8x + 5. So same result. Therefore, same conclusion.

Alternatively, if someone didn't remember the conjugate root theorem, they might try to construct the polynomial. Let me try that approach. Suppose that 4 - sqrt(11) is a root. Then p(4 - sqrt(11)) = 0. Let me assume p(x) is ax² + bx + c, with a, b, c integers. Then:

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

Let me compute (4 - sqrt(11))²:

(4 - sqrt(11))² = 16 - 8 sqrt(11) + 11 = 27 - 8 sqrt(11).

Therefore, substituting back:

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

Expanding:

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:

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 ax² + bx + c = a x² - 8a x + 5a = a(x² - 8x + 5). So again, same result. Therefore, p(x) is a multiple of x² - 8x + 5, so the ratio p(3)/p(4) is 10/11. Therefore, regardless of the value of a, the ratio is the same. Therefore, the answer is 10/11.

Alternatively, maybe someone could think that the polynomial could have other roots? But since it's quadratic, it can only have two roots. Since we already have two roots, 4 ± sqrt(11), there are no other roots. Therefore, the polynomial is uniquely determined up to a scalar multiple. Therefore, the ratio is fixed.

Therefore, I think the answer is 10/11. Let me check once more with actual numbers. Let me take p(x) = x² - 8x + 5. Then p(3) = 9 - 24 + 5 = -10, p(4) = 16 - 32 + 5 = -11. So -10/-11 = 10/11. If I take p(x) = 2x² - 16x + 10, then p(3) = 2*9 - 16*3 + 10 = 18 - 48 + 10 = -20, p(4) = 2*16 - 16*4 + 10 = 32 - 64 + 10 = -22. Then -20/-22 = 10/11. Same result. So regardless of the scalar multiple, the ratio is 10/11. Therefore, the answer is 10/11.

I think that's solid. I don't see any mistakes in my reasoning. So I think 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 polynomial $ p(x) $ must have the form:

$$
p(x) = a(x - (4 - \sqrt{11}))(x - (4 + \sqrt{11}))
$$

for some integer $ a \neq 0 $.

---

### Step 2: Simplify the Polynomial

We simplify the expression:

$$
p(x) = a(x - 4 + \sqrt{11})(x - 4 - \sqrt{11}) = a[(x - 4)^2 - (\sqrt{11})^2]
$$

$$
= a[(x - 4)^2 - 11] = a(x^2 - 8x + 16 - 11) = a(x^2 - 8x + 5)
$$

So, the polynomial is:

$$
p(x) = a(x^2 - 8x + 5)
$$

---

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

We now compute the values of $ p(3) $ and $ p(4) $:

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

---

### Step 4: Compute the Ratio

Now, compute the ratio:

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

Note that the scalar $ a $ cancels out, as long as $ a \neq 0 $, which is required for $ p(x) $ to be a quadratic polynomial.

---

### 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. That means I need to compare 3 inches to 2 feet. But since they are different units, I should convert them to the same unit first.

Alright, feet and inches. I remember that 1 foot is equal to 12 inches. So maybe I can 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 times 12 is 24. 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? Or, more formally, the fraction would be 3 inches divided by 24 inches.

Let me write that down: 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, simplifying 3/24 gives me 1/8.

Wait, so is the answer 1/8? Let me check my steps again to make sure I didn't make a mistake.

First, converting 2 feet to inches: 2 * 12 = 24 inches. Correct. Then, 3 inches divided by 24 inches. Yes, that's 3/24. Simplifying by dividing numerator and denominator by 3 gives 1/8. That seems right.

Alternatively, maybe I can approach it another way to verify. Let's think about fractions without converting units.

If I want to find what fraction 3 inches is of 2 feet, I can think of it as 3 inches over 2 feet. But since the units are different, I need to convert them. So, converting 2 feet to inches is necessary.

Alternatively, maybe convert 3 inches to feet? Let me try that.

Since 1 foot is 12 inches, 3 inches is 3/12 feet. Which simplifies to 1/4 foot. Then, the question becomes, what fraction of 2 feet is 1/4 foot?

So, that would be (1/4) / 2. Dividing by 2 is the same as multiplying by 1/2. So, (1/4) * (1/2) = 1/8.

Same answer! So that confirms it. Whether I convert the 2 feet to inches or convert the 3 inches to feet, I end up with 1/8.

Therefore, the fraction is 1/8.

Wait, just to make sure, let me think of another way. Let's think about proportions.

If 2 feet is the whole, then 3 inches is a part of that. So, how many inches are in 2 feet? 24 inches. So, 3 inches is how much of 24 inches?

So, 3 divided by 24. Which is 0.125. And 1/8 is 0.125. So that's correct.

Alternatively, if I think in terms of ratios. The ratio of 3 inches to 2 feet is 3:24, which simplifies to 1:8. So, the fraction is 1/8.

All methods lead to the same answer. Therefore, I can be confident that the answer is 1/8.

But just to make sure I didn't make any unit conversion errors. Let me check again.

1 foot = 12 inches. Therefore, 2 feet = 24 inches. 3 inches is the part. So, 3/24. Simplify. Divide numerator and denominator by 3: 1/8. Yep.

Alternatively, if I use decimal conversion. 3 inches is 0.25 feet. 0.25 divided by 2 is 0.125, which is 1/8.

Yes, that's correct.

So, no matter how I approach it, the answer is 1/8.

Therefore, 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. Since the question involves feet and inches, we'll convert feet to inches for consistency.

---

### Step 1: Convert 2 feet to inches

We know that:

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

So,

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

---

### Step 2: Express 3 inches as a fraction of 24 inches

Now, we want to find what fraction of 24 inches is 3 inches. This is a simple division:

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

---

### Step 3: Simplify the fraction

To simplify $ \frac{3}{24} $, divide both the numerator and the denominator by their greatest common divisor, which is 3:

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

---

### Final Answer

Thus, the fraction of 2 feet that is 3 inches is:

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

[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. That means I need to compare 3 inches to 2 feet. But since they are different units, I should convert them to the same unit first.

Alright, feet and inches. I remember that 1 foot is equal to 12 inches. So maybe I can 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 times 12 is 24. 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? Or, more formally, the fraction would be 3 inches divided by 24 inches.

Let me write that down: 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, simplifying 3/24 gives me 1/8.

Wait, so is the answer 1/8? Let me check my steps again to make sure I didn't make a mistake.

First, converting 2 feet to inches: 2 * 12 = 24 inches. Correct. Then, 3 inches divided by 24 inches. Yes, that's 3/24. Simplifying by dividing numerator and denominator by 3 gives 1/8. That seems right.

Alternatively, maybe I can approach it another way to verify. Let's think about fractions without converting units.

If I want to find what fraction 3 inches is of 2 feet, I can think of it as 3 inches over 2 feet. But since the units are different, I need to convert them. So, converting 2 feet to inches is necessary.

Alternatively, maybe convert 3 inches to feet? Let me try that.

Since 1 foot is 12 inches, 3 inches is 3/12 feet. Which simplifies to 1/4 foot. Then, the question becomes, what fraction of 2 feet is 1/4 foot?

So, that would be (1/4) / 2. Dividing by 2 is the same as multiplying by 1/2. So, (1/4) * (1/2) = 1/8.

Same answer! So that confirms it. Whether I convert the 2 feet to inches or convert the 3 inches to feet, I end up with 1/8.

Therefore, the fraction is 1/8.

Wait, just to make sure, let me think of another way. Let's think about proportions.

If 2 feet is the whole, then 3 inches is a part of that. So, how many inches are in 2 feet? 24 inches. So, 3 inches is how much of 24 inches?

So, 3 divided by 24. Which is 0.125. And 1/8 is 0.125. So that's correct.

Alternatively, if I think in terms of ratios. The ratio of 3 inches to 2 feet is 3:24, which simplifies to 1:8. So, the fraction is 1/8.

All methods lead to the same answer. Therefore, I can be confident that the answer is 1/8.

But just to make sure I didn't make any unit conversion errors. Let me check again.

1 foot = 12 inches. Therefore, 2 feet = 24 inches. 3 inches is the part. So, 3/24. Simplify. Divide numerator and denominator by 3: 1/8. Yep.

Alternatively, if I use decimal conversion. 3 inches is 0.25 feet. 0.25 divided by 2 is 0.125, which is 1/8.

Yes, that's correct.

So, no matter how I approach it, the answer is 1/8.

Therefore, 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. Since the question involves feet and inches, we'll convert feet to inches for consistency.

---

### Step 1: Convert 2 feet to inches

We know that:

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

So,

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

---

### Step 2: Express 3 inches as a fraction of 24 inches

Now, we want to find what fraction of 24 inches is 3 inches. This is a simple division:

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

---

### Step 3: Simplify the fraction

To simplify $ \frac{3}{24} $, divide both the numerator and the denominator by their greatest common divisor, which is 3:

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

---

### Final Answer

Thus, the fraction of 2 feet that is 3 inches is:

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

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