Explanation:
Okay, so Tom has this Mr. Potato Head, and he wants to figure out how many different wacky personalities he can make. Let me try to work through this step by step.

First, let me make sure I understand the problem correctly. The question says that a complete personality includes eyebrows, eyes, ears, lips, shoes, and optionally hair. Also, note that you can't mix and match parts, so you have to take whole sets. For example, if there are two pairs of eyebrows, you can't take one from each pair; you have to choose one entire pair.

Alright, let me list out the components that come with the Mr. Potato Head:

- 3 hairstyles
- 2 sets of eyebrows
- 1 pair of googly eyes
- 2 sets of ears
- 2 sets of lips
- 1 pair of regular shoes
- 1 bonus pair of clown shoes

So, first, I need to figure out how many choices there are for each component, and then multiply them together, since each choice is independent. But wait, there's also the option of being bald, which means that hair is optional. Let me think.

Starting with the hair. The problem says "optionally hair," so Tom can choose to have hair or not. But how does that work with the hairstyles? If he chooses to have hair, he has to pick one of the 3 hairstyles. If he doesn't, then he's bald. So for the hair component, there are 3 + 1 = 4 choices? Wait, no. Wait, actually, the number of choices for hair is either 0 (bald) or 1 of the 3 hairstyles. So that would be 3 + 1 = 4? Wait, but actually, when you have options, if something is optional, you can think of it as 1 + number of options. Because you can choose to not have it or choose one of the options. So in this case, since hair is optional, the number of choices for hair is 3 + 1 = 4? Wait, no. Wait, actually, the number of choices is 1 (no hair) plus the number of hairstyles. So that's 1 + 3 = 4. But let me check. For example, if you have 3 hairstyles, then the number of choices for hair is 3 (if you choose one) or 1 (if you choose none). So total number of choices is 3 + 1 = 4. But actually, no. Wait, in combinatorics, when you have an optional component, you can model it as multiplying by (number of choices for that component). If the component is optional, then for each component, you have (number of choices) plus 1 (for not choosing it). Wait, but in this case, hair is optional, but the hair is only available in the given number of styles. So, if you choose to have hair, you can choose one of the 3 hairstyles. If you choose not to have hair, then that's 1 option. Therefore, the total number of hair choices is 3 + 1 = 4? Wait, no. Wait, actually, if you have n options for something, the number of ways to choose that thing or not is n + 1. But in this case, hair is only optional, but the number of ways to have hair is 3, and the number of ways to not have hair is 1. Therefore, total is 3 + 1 = 4. So, yes, that's correct.

But let me make sure. Suppose there was no hair. Then, the number of hair choices would be 1 (no hair). If there are 3 hairstyles, then the number of hair choices is 1 (no hair) + 3 = 4. So, that's correct. So, hair: 4 choices.

Now, moving on to the other components. Let me list them:

- Eyebrows: 2 sets. Since you can't mix and match, you have to choose one set. So, number of choices is 2.

- Eyes: 1 pair of googly eyes. Since it's a pair, you can't choose just one eye, you have to take the whole pair. So, only 1 choice here. So, eyes: 1.

- Ears: 2 sets. Again, same as eyebrows. So, 2 choices.

- Lips: 2 sets. Similarly, 2 choices.

- Shoes: 1 pair of regular shoes and 1 bonus pair of clown shoes. So, that's 2 choices for shoes.

So, putting this all together, if we multiply all the choices together, we should get the total number of different wacky personalities.

So, total number of personalities = (choices for hair) × (choices for eyebrows) × (choices for eyes) × (choices for ears) × (choices for lips) × (choices for shoes)

So, substituting the numbers:

Total = 4 (hair) × 2 (eyebrows) × 1 (eyes) × 2 (ears) × 2 (lips) × 2 (shoes)

Let me compute that step by step.

First, compute 4 × 2 = 8

Then, 8 × 1 = 8

Then, 8 × 2 = 16

Then, 16 × 2 = 32

Then, 32 × 2 = 64

So, 64 different wacky personalities?

Wait, but hold on. Let me check again. Maybe I missed something? Let me go through each component again.

Hair: 3 hairstyles + 1 (no hair) = 4. Correct.

Eyebrows: 2 choices. Correct, since there are two different sets. You have to pick one set.

Eyes: 1 pair. Only one option. Correct.

Ears: 2 sets. Correct, two choices.

Lips: 2 sets. Correct, two choices.

Shoes: regular or clown. So two choices. Correct.

So multiplying them all together: 4×2×1×2×2×2. Let me compute that again:

Start with 4×2 = 8.

8×1 = 8.

8×2 = 16.

16×2 = 32.

32×2 = 64.

Yes, same result. So, 64? But let me think again. Wait, the problem says "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, the hair is optional. So, in other words, all other components are mandatory? Let me check.

Yes, the problem says: "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, hair is optional, others are required. Therefore, the other components (eyebrows, eyes, ears, lips, shoes) must be included. Therefore, our calculation is correct. So, for each of those, you have to choose one set (except eyes which is only one pair). So, yes, 2×1×2×2×2 = 16, multiplied by 4 for hair, gives 64.

Wait, but let me check once more. If you have to have eyebrows, eyes, ears, lips, shoes, and optionally hair. So, for each of those required parts, you have to choose one option. So:

Eyebrows: 2 options.

Eyes: 1 option.

Ears: 2 options.

Lips: 2 options.

Shoes: 2 options.

Hair: 4 options (including no hair).

Therefore, total number is 2×1×2×2×2×4 = 2×1×2×2×2×4. Let me compute that.

First, 2×1 = 2.

2×2 = 4.

4×2 = 8.

8×2 = 16.

16×4 = 64. Yep, same result.

So, 64 different personalities. Hmm. But let me think again, is there a possibility that I overcounted or undercounted?

Wait, for example, the eyes: there is only 1 pair of googly eyes. So, there's no choice there. So, only 1 way to choose eyes. Correct.

Similarly, shoes: there's regular shoes and clown shoes. So, two choices. Correct.

So, seems okay.

Wait, but here's a thought: is the hair considered as part of the personality? The problem says "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, hair is optional. Therefore, if you choose to have hair, you have to pick one of the 3 hairstyles. So, that's 3 choices. If you don't have hair, that's 1 choice. Therefore, 4 total.

So, that seems okay.

Another thing to check: are there any dependencies between the components? For example, if you have hair, does that affect anything else? The problem statement says "you cannot 'mix and match'". For example, you cannot take the left eyebrow from one pair and the right eyebrow from the other pair. But since each component is a set, you have to take the whole set. Therefore, there are no dependencies between the components. Each component is independent. Therefore, multiplying them is okay.

So, for example, choosing different hairstyles doesn't affect the number of eyebrow choices, etc. Therefore, the multiplication is correct.

Therefore, I think the answer is 64. Wait, but let me check once more.

Alternatively, maybe the question considers that the hair is optional, so you can have hair or not. But if you have hair, you have to choose one of the 3 hairstyles. Therefore, the number of hair choices is 3 (with hair) + 1 (without hair) = 4. Then, the rest are as before. So, 4×2×1×2×2×2 = 64. Therefore, 64.

But let me think again. Suppose I have to have all the other parts. So, for example, even if you are bald, you still have eyebrows, eyes, ears, lips, shoes. Therefore, the total number is indeed 4×2×1×2×2×2 = 64.

Wait, another way to think about this: if hair was mandatory, then the number of choices would be 3 (hair) × 2 × 1 × 2 × 2 × 2 = 3×16=48. But since hair is optional, we add the case where there's no hair. So, that would be 48 + 16 = 64? Wait, no. Wait, if hair is optional, then it's not just adding 16, but actually, the total is (number of choices for hair) × ... So, when hair is optional, it's (number of hair choices) × rest. So, hair choices are 4, and rest is 2×1×2×2×2= 16. 4×16=64. Alternatively, if hair was mandatory, it would be 3×rest= 3×16=48. Therefore, the difference is that when hair is optional, you have 4×16=64.

So, that seems to check out.

Alternatively, if you think about it as:

Total = (number of hair options) × (number of eyebrow options) × ... etc.

But since hair is optional, the number of hair options is 4 as above.

So, I think this is correct. Therefore, 64 different personalities.

But let me check once more for errors. Sometimes, when problems say "optional", you have to be careful if there's a possibility of zero or more, but in this case, it's either zero or one. Since hair is a single component, and you can't have multiple hairstyles. So, yeah, 3 + 1 = 4.

Alternatively, maybe the problem considers that "optionally hair" means that hair is optional, but if you have hair, you have to choose one of the 3. So, that's 3 + 1. So, that's correct.

Therefore, 4×2×1×2×2×2 = 64. Therefore, the answer is 64.

Wait, but let me check again if perhaps the eyes are considered as optional? Wait, the problem says "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, eyes are included as a required part. Therefore, there's only one pair of eyes, so only one choice. So, that's correct.

Another check: let's think about each component.

Hair: 4 choices.

Eyebrows: 2 choices.

Eyes: 1 choice.

Ears: 2 choices.

Lips: 2 choices.

Shoes: 2 choices.

So, total combinations: 4*2*1*2*2*2 = 4*2*1*8 = 4*16 = 64. Yes. So, that's 64.

I think that's solid. Therefore, I think the answer is 64.

**Final Answer**
The number of different wacky personalities Tom can come up with is \boxed{64}.

To determine the number of different **wacky personalities** Tom can create with his Mr. Potato Head, we need to consider all the components provided, and how many choices exist for each component.

We are told that a complete personality includes the following **mandatory** elements:

- **Eyebrows**
- **Eyes**
- **Ears**
- **Lips**
- **Shoes**

And **hair** is **optional**. Additionally, the problem indicates that all parts are taken as **whole sets**, which means you **cannot mix and match** parts from different sets (e.g., choose one eyebrow from one set and one from another).

---

### Step 1: Determine the Number of Choices for Each Component

- **Hair**: There are 3 **hairstyles**, and the option to be **bald** (i.e., have no hair) is also available.
  → Total choices = 3 (hairstyles) + 1 (no hair) = **4**

- **Eyebrows**: 2 sets → choose **one set**
  → **2 choices**

- **Eyes**: Only **1 pair** of googly eyes
  → **1 choice**

- **Ears**: 2 sets → choose **one set**
  → **2 choices**

- **Lips**: 2 sets → choose **one set**
  → **2 choices**

- **Shoes**: Regular shoes and clown shoes → **2 choices**

---

### Step 2: Compute the Total Number of Combinations

Since each component is independent, the total number of unique combinations is the **product** of the number of choices for each component:

$$
\text{Total} = 4 \ (\text{hair}) \times 2 \ (\text{eyebrows}) \times 1 \ (\text{eyes}) \times 2 \ (\text{ears}) \times 2 \ (\text{lips}) \times 2 \ (\text{shoes})
$$

We compute step by step:

- $ 4 \times 2 = 8 $
- $ 8 \times 1 = 8 $
- $ 8 \times 2 = 16 $
- $ 16 \times 2 = 32 $
- $ 32 \times 2 = 64 $

---

### Final Answer

The total number of different **wacky personalities** Tom can create is:

$$
\boxed{64}
$$

[Assistant]
Explanation:
Okay, so Tom has this Mr. Potato Head, and he wants to figure out how many different wacky personalities he can make. Let me try to work through this step by step.

First, let me make sure I understand the problem correctly. The question says that a complete personality includes eyebrows, eyes, ears, lips, shoes, and optionally hair. Also, note that you can't mix and match parts, so you have to take whole sets. For example, if there are two pairs of eyebrows, you can't take one from each pair; you have to choose one entire pair.

Alright, let me list out the components that come with the Mr. Potato Head:

- 3 hairstyles
- 2 sets of eyebrows
- 1 pair of googly eyes
- 2 sets of ears
- 2 sets of lips
- 1 pair of regular shoes
- 1 bonus pair of clown shoes

So, first, I need to figure out how many choices there are for each component, and then multiply them together, since each choice is independent. But wait, there's also the option of being bald, which means that hair is optional. Let me think.

Starting with the hair. The problem says "optionally hair," so Tom can choose to have hair or not. But how does that work with the hairstyles? If he chooses to have hair, he has to pick one of the 3 hairstyles. If he doesn't, then he's bald. So for the hair component, there are 3 + 1 = 4 choices? Wait, no. Wait, actually, the number of choices for hair is either 0 (bald) or 1 of the 3 hairstyles. So that would be 3 + 1 = 4? Wait, but actually, when you have options, if something is optional, you can think of it as 1 + number of options. Because you can choose to not have it or choose one of the options. So in this case, since hair is optional, the number of choices for hair is 3 + 1 = 4? Wait, no. Wait, actually, the number of choices is 1 (no hair) plus the number of hairstyles. So that's 1 + 3 = 4. But let me check. For example, if you have 3 hairstyles, then the number of choices for hair is 3 (if you choose one) or 1 (if you choose none). So total number of choices is 3 + 1 = 4. But actually, no. Wait, in combinatorics, when you have an optional component, you can model it as multiplying by (number of choices for that component). If the component is optional, then for each component, you have (number of choices) plus 1 (for not choosing it). Wait, but in this case, hair is optional, but the hair is only available in the given number of styles. So, if you choose to have hair, you can choose one of the 3 hairstyles. If you choose not to have hair, then that's 1 option. Therefore, the total number of hair choices is 3 + 1 = 4? Wait, no. Wait, actually, if you have n options for something, the number of ways to choose that thing or not is n + 1. But in this case, hair is only optional, but the number of ways to have hair is 3, and the number of ways to not have hair is 1. Therefore, total is 3 + 1 = 4. So, yes, that's correct.

But let me make sure. Suppose there was no hair. Then, the number of hair choices would be 1 (no hair). If there are 3 hairstyles, then the number of hair choices is 1 (no hair) + 3 = 4. So, that's correct. So, hair: 4 choices.

Now, moving on to the other components. Let me list them:

- Eyebrows: 2 sets. Since you can't mix and match, you have to choose one set. So, number of choices is 2.

- Eyes: 1 pair of googly eyes. Since it's a pair, you can't choose just one eye, you have to take the whole pair. So, only 1 choice here. So, eyes: 1.

- Ears: 2 sets. Again, same as eyebrows. So, 2 choices.

- Lips: 2 sets. Similarly, 2 choices.

- Shoes: 1 pair of regular shoes and 1 bonus pair of clown shoes. So, that's 2 choices for shoes.

So, putting this all together, if we multiply all the choices together, we should get the total number of different wacky personalities.

So, total number of personalities = (choices for hair) × (choices for eyebrows) × (choices for eyes) × (choices for ears) × (choices for lips) × (choices for shoes)

So, substituting the numbers:

Total = 4 (hair) × 2 (eyebrows) × 1 (eyes) × 2 (ears) × 2 (lips) × 2 (shoes)

Let me compute that step by step.

First, compute 4 × 2 = 8

Then, 8 × 1 = 8

Then, 8 × 2 = 16

Then, 16 × 2 = 32

Then, 32 × 2 = 64

So, 64 different wacky personalities?

Wait, but hold on. Let me check again. Maybe I missed something? Let me go through each component again.

Hair: 3 hairstyles + 1 (no hair) = 4. Correct.

Eyebrows: 2 choices. Correct, since there are two different sets. You have to pick one set.

Eyes: 1 pair. Only one option. Correct.

Ears: 2 sets. Correct, two choices.

Lips: 2 sets. Correct, two choices.

Shoes: regular or clown. So two choices. Correct.

So multiplying them all together: 4×2×1×2×2×2. Let me compute that again:

Start with 4×2 = 8.

8×1 = 8.

8×2 = 16.

16×2 = 32.

32×2 = 64.

Yes, same result. So, 64? But let me think again. Wait, the problem says "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, the hair is optional. So, in other words, all other components are mandatory? Let me check.

Yes, the problem says: "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, hair is optional, others are required. Therefore, the other components (eyebrows, eyes, ears, lips, shoes) must be included. Therefore, our calculation is correct. So, for each of those, you have to choose one set (except eyes which is only one pair). So, yes, 2×1×2×2×2 = 16, multiplied by 4 for hair, gives 64.

Wait, but let me check once more. If you have to have eyebrows, eyes, ears, lips, shoes, and optionally hair. So, for each of those required parts, you have to choose one option. So:

Eyebrows: 2 options.

Eyes: 1 option.

Ears: 2 options.

Lips: 2 options.

Shoes: 2 options.

Hair: 4 options (including no hair).

Therefore, total number is 2×1×2×2×2×4 = 2×1×2×2×2×4. Let me compute that.

First, 2×1 = 2.

2×2 = 4.

4×2 = 8.

8×2 = 16.

16×4 = 64. Yep, same result.

So, 64 different personalities. Hmm. But let me think again, is there a possibility that I overcounted or undercounted?

Wait, for example, the eyes: there is only 1 pair of googly eyes. So, there's no choice there. So, only 1 way to choose eyes. Correct.

Similarly, shoes: there's regular shoes and clown shoes. So, two choices. Correct.

So, seems okay.

Wait, but here's a thought: is the hair considered as part of the personality? The problem says "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, hair is optional. Therefore, if you choose to have hair, you have to pick one of the 3 hairstyles. So, that's 3 choices. If you don't have hair, that's 1 choice. Therefore, 4 total.

So, that seems okay.

Another thing to check: are there any dependencies between the components? For example, if you have hair, does that affect anything else? The problem statement says "you cannot 'mix and match'". For example, you cannot take the left eyebrow from one pair and the right eyebrow from the other pair. But since each component is a set, you have to take the whole set. Therefore, there are no dependencies between the components. Each component is independent. Therefore, multiplying them is okay.

So, for example, choosing different hairstyles doesn't affect the number of eyebrow choices, etc. Therefore, the multiplication is correct.

Therefore, I think the answer is 64. Wait, but let me check once more.

Alternatively, maybe the question considers that the hair is optional, so you can have hair or not. But if you have hair, you have to choose one of the 3 hairstyles. Therefore, the number of hair choices is 3 (with hair) + 1 (without hair) = 4. Then, the rest are as before. So, 4×2×1×2×2×2 = 64. Therefore, 64.

But let me think again. Suppose I have to have all the other parts. So, for example, even if you are bald, you still have eyebrows, eyes, ears, lips, shoes. Therefore, the total number is indeed 4×2×1×2×2×2 = 64.

Wait, another way to think about this: if hair was mandatory, then the number of choices would be 3 (hair) × 2 × 1 × 2 × 2 × 2 = 3×16=48. But since hair is optional, we add the case where there's no hair. So, that would be 48 + 16 = 64? Wait, no. Wait, if hair is optional, then it's not just adding 16, but actually, the total is (number of choices for hair) × ... So, when hair is optional, it's (number of hair choices) × rest. So, hair choices are 4, and rest is 2×1×2×2×2= 16. 4×16=64. Alternatively, if hair was mandatory, it would be 3×rest= 3×16=48. Therefore, the difference is that when hair is optional, you have 4×16=64.

So, that seems to check out.

Alternatively, if you think about it as:

Total = (number of hair options) × (number of eyebrow options) × ... etc.

But since hair is optional, the number of hair options is 4 as above.

So, I think this is correct. Therefore, 64 different personalities.

But let me check once more for errors. Sometimes, when problems say "optional", you have to be careful if there's a possibility of zero or more, but in this case, it's either zero or one. Since hair is a single component, and you can't have multiple hairstyles. So, yeah, 3 + 1 = 4.

Alternatively, maybe the problem considers that "optionally hair" means that hair is optional, but if you have hair, you have to choose one of the 3. So, that's 3 + 1. So, that's correct.

Therefore, 4×2×1×2×2×2 = 64. Therefore, the answer is 64.

Wait, but let me check again if perhaps the eyes are considered as optional? Wait, the problem says "a complete Mr. Potato Head personality includes eyebrows, eyes, ears, lips, shoes and optionally hair". So, eyes are included as a required part. Therefore, there's only one pair of eyes, so only one choice. So, that's correct.

Another check: let's think about each component.

Hair: 4 choices.

Eyebrows: 2 choices.

Eyes: 1 choice.

Ears: 2 choices.

Lips: 2 choices.

Shoes: 2 choices.

So, total combinations: 4*2*1*2*2*2 = 4*2*1*8 = 4*16 = 64. Yes. So, that's 64.

I think that's solid. Therefore, I think the answer is 64.

**Final Answer**
The number of different wacky personalities Tom can come up with is \boxed{64}.

To determine the number of different **wacky personalities** Tom can create with his Mr. Potato Head, we need to consider all the components provided, and how many choices exist for each component.

We are told that a complete personality includes the following **mandatory** elements:

- **Eyebrows**
- **Eyes**
- **Ears**
- **Lips**
- **Shoes**

And **hair** is **optional**. Additionally, the problem indicates that all parts are taken as **whole sets**, which means you **cannot mix and match** parts from different sets (e.g., choose one eyebrow from one set and one from another).

---

### Step 1: Determine the Number of Choices for Each Component

- **Hair**: There are 3 **hairstyles**, and the option to be **bald** (i.e., have no hair) is also available.
  → Total choices = 3 (hairstyles) + 1 (no hair) = **4**

- **Eyebrows**: 2 sets → choose **one set**
  → **2 choices**

- **Eyes**: Only **1 pair** of googly eyes
  → **1 choice**

- **Ears**: 2 sets → choose **one set**
  → **2 choices**

- **Lips**: 2 sets → choose **one set**
  → **2 choices**

- **Shoes**: Regular shoes and clown shoes → **2 choices**

---

### Step 2: Compute the Total Number of Combinations

Since each component is independent, the total number of unique combinations is the **product** of the number of choices for each component:

$$
\text{Total} = 4 \ (\text{hair}) \times 2 \ (\text{eyebrows}) \times 1 \ (\text{eyes}) \times 2 \ (\text{ears}) \times 2 \ (\text{lips}) \times 2 \ (\text{shoes})
$$

We compute step by step:

- $ 4 \times 2 = 8 $
- $ 8 \times 1 = 8 $
- $ 8 \times 2 = 16 $
- $ 16 \times 2 = 32 $
- $ 32 \times 2 = 64 $

---

### Final Answer

The total number of different **wacky personalities** Tom can create is:

$$
\boxed{64}
$$

Solution: There are 2 choices of eyebrows, 1 choice for eyes, 2 choices for ears, 2 choices for lips, 2 choices for shoes, and 4 choices for hair (3 choices of hairstyles, or bald), and each can be picked independent of the others, so the number of combinations is the product of those, $2\cdot 1\cdot 2\cdot 2\cdot 2\cdot 4 = \boxed{64}$.


Explanation:
Okay, so I need to figure out how to express $(\sqrt{2} + \sqrt{3})^3$ in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$ where $a$, $b$, and $c$ are integers. Then, I have to find the sum $a + b + c$. Alright, let me think about how to approach this.

First, I remember that when you have something like $(a + b)^3$, you can expand it using the binomial theorem. The binomial theorem says that $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. Maybe I can apply that here. Let me try that.

Let me denote $a = \sqrt{2}$ and $b = \sqrt{3}$. Then, $(\sqrt{2} + \sqrt{3})^3$ should be equal to:

$$
(\sqrt{2})^3 + 3(\sqrt{2})^2(\sqrt{3}) + 3(\sqrt{2})(\sqrt{3})^2 + (\sqrt{3})^3
$$

Let me compute each term one by one. Starting with the first term:

1. $(\sqrt{2})^3$: $\sqrt{2} \times \sqrt{2} \times \sqrt{2}$. Since $\sqrt{2} \times \sqrt{2} = 2$, so that term becomes $2 \times \sqrt{2} = 2\sqrt{2}$.

2. The second term is $3(\sqrt{2})^2(\sqrt{3})$. Let me compute $(\sqrt{2})^2$ first, which is 2. Then multiply by $\sqrt{3}$: $2 \times \sqrt{3} = 2\sqrt{3}$. Then multiply by 3: $3 \times 2\sqrt{3} = 6\sqrt{3}$.

3. The third term is $3(\sqrt{2})(\sqrt{3})^2$. Let me compute $(\sqrt{3})^2$ first, which is 3. Then multiply by $\sqrt{2}$: $3 \times \sqrt{2} = 3\sqrt{2}$. Then multiply by 3: $3 \times 3\sqrt{2} = 9\sqrt{2}$.

4. The fourth term is $(\sqrt{3})^3$: $\sqrt{3} \times \sqrt{3} \times \sqrt{3}$. Again, $\sqrt{3} \times \sqrt{3} = 3$, so that term becomes $3 \times \sqrt{3} = 3\sqrt{3}$.

Now, let me put all these terms together:

First term: $2\sqrt{2}$

Second term: $6\sqrt{3}$

Third term: $9\sqrt{2}$

Fourth term: $3\sqrt{3}$

So adding them up:

For $\sqrt{2}$ terms: $2\sqrt{2} + 9\sqrt{2} = (2 + 9)\sqrt{2} = 11\sqrt{2}$

For $\sqrt{3}$ terms: $6\sqrt{3} + 3\sqrt{3} = (6 + 3)\sqrt{3} = 9\sqrt{3}$

Wait, but the problem states that the expression should be in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$. However, right now I only have terms with $\sqrt{2}$ and $\sqrt{3}$. There's no $\sqrt{6}$ term. Hmm, did I do something wrong?

Wait a minute, maybe I made a mistake. Let me check my calculations again.

Starting over, expanding $(\sqrt{2} + \sqrt{3})^3$:

First term: $(\sqrt{2})^3 = 2\sqrt{2}$

Second term: $3(\sqrt{2})^2(\sqrt{3})$. $(\sqrt{2})^2 = 2$, so 3 * 2 * $\sqrt{3}$ = 6$\sqrt{3}$, that seems right.

Third term: $3(\sqrt{2})(\sqrt{3})^2$. $(\sqrt{3})^2 = 3, so 3 * $\sqrt{2}$ * 3 = 9$\sqrt{2}$, that seems okay.

Fourth term: $(\sqrt{3})^3 = 3\sqrt{3}$

So adding them up: 2√2 + 6√3 + 9√2 + 3√3. Which is (2 + 9)√2 + (6 + 3)√3 = 11√2 + 9√3. Wait, so there is no √6 term? But the problem says it can be written as a√2 + b√3 + c√6. So maybe there is a mistake here?

Wait, maybe I need to check my expansion again. Wait, perhaps I missed a cross term that involves √6?

Wait, hold on. Let me think again. Is there a different way to expand $(\sqrt{2} + \sqrt{3})^3$? Maybe using another method?

Alternatively, maybe I can compute $(\sqrt{2} + \sqrt{3})^3$ by first squaring $(\sqrt{2} + \sqrt{3})$ and then multiplying by $(\sqrt{2} + \sqrt{3})$ again?

Let me try that approach. Let me compute $(\sqrt{2} + \sqrt{3})^2$ first.

$$
(\sqrt{2} + \sqrt{3})^2 = (\sqrt{2})^2 + 2\sqrt{2}\sqrt{3} + (\sqrt{3})^2 = 2 + 2\sqrt{6} + 3 = 5 + 2\sqrt{6}
$$

Okay, so that's the square. Then, to get the cube, I need to multiply that by $(\sqrt{2} + \sqrt{3})$:

$$
(5 + 2\sqrt{6})(\sqrt{2} + \sqrt{3})
$$

Let me expand this:

First, multiply 5 by $\sqrt{2}$: $5\sqrt{2}$

Then 5 times $\sqrt{3}$: $5\sqrt{3}$

Then $2\sqrt{6} \times \sqrt{2}$: Let me compute $\sqrt{6} \times \sqrt{2}$. That's $\sqrt{12}$, which simplifies to $2\sqrt{3}$. So 2 times that is $4\sqrt{3}$.

Then $2\sqrt{6} \times \sqrt{3}$: $\sqrt{6} \times \sqrt{3} = \sqrt{18} = 3\sqrt{2}$. So 2 times that is $6\sqrt{2}$.

So putting all those terms together:

From the first multiplication: 5√2 + 5√3

From the second multiplication: 4√3 + 6√2

Adding them all up:

For √2 terms: 5√2 + 6√2 = 11√2

For √3 terms: 5√3 + 4√3 = 9√3

So again, same result: 11√2 + 9√3. But there is no √6 term here. However, the problem says it can be written as a√2 + b√3 + c√6. So maybe there is a mistake in my calculation, or maybe c is zero?

Wait, the problem says "can be written in the form a√2 + b√3 + c√6, where a, b, and c are integers." So maybe c is zero? But why would they include that term? Maybe there's an error in my calculation?

Wait, but let me check again. Let me compute $(\sqrt{2} + \sqrt{3})^3$ numerically, and also compute my answer numerically to check.

First, let's compute the original expression numerically. Let me calculate sqrt(2) ≈ 1.4142, sqrt(3) ≈ 1.732.

So sqrt(2) + sqrt(3) ≈ 1.4142 + 1.732 ≈ 3.1462.

Then, cube that: 3.1462^3. Let me compute that:

First, 3^3 is 27. Let me compute 3.1462^3:

3.1462 * 3.1462 = ?

Let me compute 3 * 3.1462 = 9.4386, 0.1462 * 3.1462 ≈ approximately 0.1462 * 3 ≈ 0.4386, plus 0.1462 * 0.1462 ≈ ~0.0214. So total approx 0.4386 + 0.0214 ≈ 0.46. So total approx 9.4386 + 0.46 ≈ 9.8986. Then multiply by 3.1462 again:

9.8986 * 3.1462 ≈ Let's approximate:

10 * 3.1462 = 31.462

Subtract 0.1014 * 3.1462 ≈ 0.319

So approx 31.462 - 0.319 ≈ 31.143.

So approx 31.14.

Now let me compute my expression: 11√2 + 9√3.

Compute 11√2 ≈ 11 * 1.4142 ≈ 15.5562

Compute 9√3 ≈ 9 * 1.732 ≈ 15.588

Adding them together: 15.5562 + 15.588 ≈ 31.1442. Which is very close to the earlier approximation of 31.14. So that checks out. So my calculation of 11√2 + 9√3 seems correct.

But the problem says it can be written in the form a√2 + b√3 + c√6. So why is there no √6 term? Maybe the problem is written in a way that allows c to be zero? So maybe the answer is a = 11, b = 9, c = 0, so a + b + c = 20?

But why would the problem mention that form if c is zero? Maybe I made a mistake in the problem statement?

Wait, let me check again. Wait, the original problem says: "The number $(\sqrt{2}+\sqrt{3})^3$ can be written in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers. What is $a+b+c$?" So maybe they just included the √6 term as a possible term, but it's allowed for c to be zero. So maybe the answer is 11 + 9 + 0 = 20? But let me check again.

Alternatively, perhaps my expansion is missing something? Let me think. Let me try to expand $(\sqrt{2} + \sqrt{3})^3$ again step by step.

Alternatively, perhaps I should use the binomial coefficients again. Let me write the expansion again:

$(\sqrt{2} + \sqrt{3})^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(\sqrt{3}) + 3(\sqrt{2})(\sqrt{3})^2 + (\sqrt{3})^3$.

Compute each term:

First term: $(\sqrt{2})^3 = 2^{3/2} = 2 \sqrt{2}$

Second term: $3 \times (\sqrt{2})^2 \times \sqrt{3} = 3 \times 2 \times \sqrt{3} = 6 \sqrt{3}$

Third term: $3 \times \sqrt{2} \times (\sqrt{3})^2 = 3 \times \sqrt{2} \times 3 = 9 \sqrt{2}$

Fourth term: $(\sqrt{3})^3 = 3^{3/2} = 3 \sqrt{3}$

Adding all together:

First term: 2√2

Second term: 6√3

Third term: 9√2

Fourth term: 3√3

So adding the like terms:

For √2: 2√2 + 9√2 = 11√2

For √3: 6√3 + 3√3 = 9√3

No √6 terms here. So unless there is an error in my calculation, the expression is 11√2 + 9√3 + 0√6. So that would make c = 0, so a + b + c = 11 + 9 + 0 = 20. Is that the answer?

But maybe there's another approach where there is a √6 term? Let me think. Maybe I need to consider another expansion?

Wait, let me check if there's a different way to write the expression. For example, maybe combining terms?

Alternatively, could there be an error in the problem statement? Or perhaps I need to rationalize or something else? Wait, but the problem says that it can be written in that form, so maybe c is zero. Maybe the answer is 20?

Alternatively, maybe I made a mistake in my numerical calculation? Let me check with more precise numbers.

Let me calculate $(\sqrt{2} + \sqrt{3})^3$ more accurately.

First, exact value. Wait, if I have 11√2 + 9√3, then let me compute that with more precise decimal approx. Let me use more accurate values:

sqrt(2) ≈ 1.41421356

sqrt(3) ≈ 1.73205080757

So 11√2 ≈ 11 * 1.41421356 ≈ 15.55634916

9√3 ≈ 9 * 1.73205080757 ≈ 15.588457268

Adding those two gives ≈ 15.55634916 + 15.588457268 ≈ 31.14480643

Now let's compute the original expression numerically:

sqrt(2) + sqrt(3) ≈ 1.41421356 + 1.73205080757 ≈ 3.14626436757

Now, cube that:

First, square it: (3.14626436757)^2 ≈ Let me compute that:

First, 3^2 = 9, 0.14626436757^2 ≈ approx 0.0214, and cross terms:

Using (a + b)^2 = a^2 + 2ab + b^2 where a = 3, b = 0.14626436757.

So that's 9 + 2*3*0.14626436757 + (0.14626436757)^2 ≈ 9 + 0.8775862054 + 0.0214 ≈ 9.8989862054

Now multiply that by 3.14626436757:

So 9.8989862054 * 3.14626436757 ≈ Let me compute this more accurately.

Let me note that 9.8989862054 * 3 = 29.6969586162

Then, 9.8989862054 * 0.14626436757 ≈ ?

First, approximate 9.8989862054 * 0.1 = 0.98989862054

9.8989862054 * 0.04 = 0.395959448216

9.8989862054 * 0.00626436757 ≈ Let's approximate:

0.006 * 9.8989862054 ≈ 0.05939391723

Adding up: 0.98989862054 + 0.395959448216 ≈ 1.38585806876 + 0.05939391723 ≈ 1.445251986

So total approximate value is 29.6969586162 + 1.445251986 ≈ 31.1422106

Which is very close to the previous calculation of 31.14480643. The slight difference is due to rounding errors. Therefore, my expression 11√2 + 9√3 is accurate.

So, since there's no √6 term, c must be zero, so a + b + c = 11 + 9 + 0 = 20. Therefore, the answer is 20.

But wait, the problem says "can be written in the form...", so maybe there's another way to express it with a √6 term? Let me think. Suppose that maybe when expanding, there is a term with √6?

Wait, let me think again. If I consider that maybe in my expansion, I missed a cross term? Wait, but in the expansion, when we expand $(a + b)^3$, there are only four terms, as per binomial theorem. When I expanded using the square first, I only got terms with √2 and √3, no √6 terms. Wait, unless there is an error in my calculation?

Wait, let me try expanding $(\sqrt{2} + \sqrt{3})^3$ using another method. Let me consider multiplying step by step:

Let me compute $(\sqrt{2} + \sqrt{3})^3 = (\sqrt{2} + \sqrt{3}) \times (\sqrt{2} + \sqrt{3}) \times (\sqrt{2} + \sqrt{3})$

First compute the first two factors: $(\sqrt{2} + \sqrt{3})^2 = 5 + 2\sqrt{6}$ as before.

Now multiply that by $(\sqrt{2} + \sqrt{3})$:

(5 + 2√6)(√2 + √3) = 5√2 + 5√3 + 2√6√2 + 2√6√3

Compute the terms:

First term: 5√2

Second term: 5√3

Third term: 2√6 * √2 = 2√(6*2) = 2√12 = 2*2√3 = 4√3

Fourth term: 2√6 * √3 = 2√(6*3) = 2√18 = 2*3√2 = 6√2

So combining all terms:

5√2 + 5√3 + 4√3 + 6√2 = (5√2 + 6√2) + (5√3 + 4√3) = 11√2 + 9√3

Same result again. So there's no term with √6. Therefore, c is zero. Therefore, the answer is 11 + 9 + 0 = 20.

But why does the problem mention the form including c√6? Maybe it's possible that in some other expansion, there might be a √6 term? Or maybe it's just including that term as a possibility, even if c=0? Maybe the problem expects the answer to have c=0? So then the answer is 20. But maybe I need to confirm once again.

Alternatively, maybe I made a mistake in the problem statement? Wait, the problem says "the number can be written in the form a√2 + b√3 + c√6". So even if c is zero, that's allowed. So, in that case, the answer is 11 + 9 + 0 = 20. Therefore, I think that's the answer. Maybe the problem is designed to have c=0, but just include that term in the form for generality.

Alternatively, let me try to think if there is another approach where there is a √6 term. For example, maybe if we consider that when expanding, there is a term that combines to √6?

Wait, let me think. If I consider that $(\sqrt{2} + \sqrt{3})^3$ might have terms that can be combined with √6. For example, suppose we have some terms that can be expressed as a multiple of √6?

Wait, but in my previous steps, all terms were either √2 or √3. Unless I made a mistake in simplifying the terms. Let me check the third term again:

Third term when I did the expansion was 2√6 * √2. Wait, no, when I multiplied (5 + 2√6)(√2 + √3), the third term was 2√6 * √2. Let me compute that again:

√6 * √2 = √(6*2) = √12 = 2√3. So 2√6 * √2 is 2 * 2√3 = 4√3. Correct. Similarly, 2√6 * √3 is √(6*3) = √18 = 3√2, so times 2 is 6√2. Correct.

So all terms are accounted for, and there is no √6 term. Therefore, conclusion is that c=0.

Therefore, the answer is 11 + 9 + 0 = 20. Therefore, I think that is the answer.

Wait, but I'm a bit confused because the problem specifically includes the √6 term in the form. Maybe there is an alternative way to write this expression with √6? Let me think again. Suppose that I didn't use the binomial theorem, but instead tried to express it as something else. Wait, but I think that my calculations are correct.

Alternatively, maybe the problem had a typo? But assuming that the problem is correct, then the answer is 20.

Alternatively, maybe there is a miscalculation in my part? Let me check once again.

Let me try to compute $(\sqrt{2} + \sqrt{3})^3$ using another approach. Let me denote $x = \sqrt{2}$ and $y = \sqrt{3}$, so that the expression is $(x + y)^3$. Expanding this:

$x^3 + 3x^2y + 3xy^2 + y^3$. As before.

Compute each term:

$x^3 = (\sqrt{2})^3 = 2\sqrt{2}$

$3x^2y = 3*(\sqrt{2})^2*\sqrt{3} = 3*2*\sqrt{3} = 6\sqrt{3}$

$3xy^2 = 3*\sqrt{2}*(\sqrt{3})^2 = 3*\sqrt{2}*3 = 9\sqrt{2}$

$y^3 = (\sqrt{3})^3 = 3\sqrt{3}$

Adding all terms:

2√2 + 6√3 + 9√2 + 3√3 = (2 + 9)√2 + (6 + 3)√3 = 11√2 + 9√3. Same result. So there's no √6 term. So if the problem says that it can be written in that form, then c = 0. Thus, a + b + c = 11 + 9 + 0 = 20.

Therefore, I think the answer is 20. Maybe the problem includes the √6 term to trick students into thinking there's a cross term, but in reality, there isn't. So the answer is 20. Let me check online if possible? Wait, I can't check, but since my calculations are consistent across multiple methods and numerical verification aligns, I think that is correct.

Thus, the answer should be 20.

**Final Answer**
\boxed{20}

To evaluate the expression $(\sqrt{2} + \sqrt{3})^3$ and express it in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers, we begin by applying the **binomial theorem**:

$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$

Let us define:
- $a = \sqrt{2}$
- $b = \sqrt{3}$

Then the expression becomes:

$$
(\sqrt{2} + \sqrt{3})^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(\sqrt{3}) + 3(\sqrt{2})(\sqrt{3})^2 + (\sqrt{3})^3
$$

Now compute each term individually:

1. $(\sqrt{2})^3 = \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = 2\sqrt{2}$

2. $3(\sqrt{2})^2(\sqrt{3}) = 3 \cdot 2 \cdot \sqrt{3} = 6\sqrt{3}$

3. $3(\sqrt{2})(\sqrt{3})^2 = 3 \cdot \sqrt{2} \cdot 3 = 9\sqrt{2}$

4. $(\sqrt{3})^3 = \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = 3\sqrt{3}$

Now add all the terms together:

- $\sqrt{2}$ terms: $2\sqrt{2} + 9\sqrt{2} = 11\sqrt{2}$
- $\sqrt{3}$ terms: $6\sqrt{3} + 3\sqrt{3} = 9\sqrt{3}$

So, the expression simplifies to:

$$
(\sqrt{2} + \sqrt{3})^3 = 11\sqrt{2} + 9\sqrt{3}
$$

---

This is of the form:

$$
a\sqrt{2} + b\sqrt{3} + c\sqrt{6}
$$

where:
- $a = 11$
- $b = 9$
- $c = 0$

Thus, the sum $a + b + c = 11 + 9 + 0 = \boxed{20}$

---

**Final Answer:** $\boxed{20}$

[Assistant]
Explanation:
Okay, so I need to figure out how to express $(\sqrt{2} + \sqrt{3})^3$ in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$ where $a$, $b$, and $c$ are integers. Then, I have to find the sum $a + b + c$. Alright, let me think about how to approach this.

First, I remember that when you have something like $(a + b)^3$, you can expand it using the binomial theorem. The binomial theorem says that $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. Maybe I can apply that here. Let me try that.

Let me denote $a = \sqrt{2}$ and $b = \sqrt{3}$. Then, $(\sqrt{2} + \sqrt{3})^3$ should be equal to:

$$
(\sqrt{2})^3 + 3(\sqrt{2})^2(\sqrt{3}) + 3(\sqrt{2})(\sqrt{3})^2 + (\sqrt{3})^3
$$

Let me compute each term one by one. Starting with the first term:

1. $(\sqrt{2})^3$: $\sqrt{2} \times \sqrt{2} \times \sqrt{2}$. Since $\sqrt{2} \times \sqrt{2} = 2$, so that term becomes $2 \times \sqrt{2} = 2\sqrt{2}$.

2. The second term is $3(\sqrt{2})^2(\sqrt{3})$. Let me compute $(\sqrt{2})^2$ first, which is 2. Then multiply by $\sqrt{3}$: $2 \times \sqrt{3} = 2\sqrt{3}$. Then multiply by 3: $3 \times 2\sqrt{3} = 6\sqrt{3}$.

3. The third term is $3(\sqrt{2})(\sqrt{3})^2$. Let me compute $(\sqrt{3})^2$ first, which is 3. Then multiply by $\sqrt{2}$: $3 \times \sqrt{2} = 3\sqrt{2}$. Then multiply by 3: $3 \times 3\sqrt{2} = 9\sqrt{2}$.

4. The fourth term is $(\sqrt{3})^3$: $\sqrt{3} \times \sqrt{3} \times \sqrt{3}$. Again, $\sqrt{3} \times \sqrt{3} = 3$, so that term becomes $3 \times \sqrt{3} = 3\sqrt{3}$.

Now, let me put all these terms together:

First term: $2\sqrt{2}$

Second term: $6\sqrt{3}$

Third term: $9\sqrt{2}$

Fourth term: $3\sqrt{3}$

So adding them up:

For $\sqrt{2}$ terms: $2\sqrt{2} + 9\sqrt{2} = (2 + 9)\sqrt{2} = 11\sqrt{2}$

For $\sqrt{3}$ terms: $6\sqrt{3} + 3\sqrt{3} = (6 + 3)\sqrt{3} = 9\sqrt{3}$

Wait, but the problem states that the expression should be in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$. However, right now I only have terms with $\sqrt{2}$ and $\sqrt{3}$. There's no $\sqrt{6}$ term. Hmm, did I do something wrong?

Wait a minute, maybe I made a mistake. Let me check my calculations again.

Starting over, expanding $(\sqrt{2} + \sqrt{3})^3$:

First term: $(\sqrt{2})^3 = 2\sqrt{2}$

Second term: $3(\sqrt{2})^2(\sqrt{3})$. $(\sqrt{2})^2 = 2$, so 3 * 2 * $\sqrt{3}$ = 6$\sqrt{3}$, that seems right.

Third term: $3(\sqrt{2})(\sqrt{3})^2$. $(\sqrt{3})^2 = 3, so 3 * $\sqrt{2}$ * 3 = 9$\sqrt{2}$, that seems okay.

Fourth term: $(\sqrt{3})^3 = 3\sqrt{3}$

So adding them up: 2√2 + 6√3 + 9√2 + 3√3. Which is (2 + 9)√2 + (6 + 3)√3 = 11√2 + 9√3. Wait, so there is no √6 term? But the problem says it can be written as a√2 + b√3 + c√6. So maybe there is a mistake here?

Wait, maybe I need to check my expansion again. Wait, perhaps I missed a cross term that involves √6?

Wait, hold on. Let me think again. Is there a different way to expand $(\sqrt{2} + \sqrt{3})^3$? Maybe using another method?

Alternatively, maybe I can compute $(\sqrt{2} + \sqrt{3})^3$ by first squaring $(\sqrt{2} + \sqrt{3})$ and then multiplying by $(\sqrt{2} + \sqrt{3})$ again?

Let me try that approach. Let me compute $(\sqrt{2} + \sqrt{3})^2$ first.

$$
(\sqrt{2} + \sqrt{3})^2 = (\sqrt{2})^2 + 2\sqrt{2}\sqrt{3} + (\sqrt{3})^2 = 2 + 2\sqrt{6} + 3 = 5 + 2\sqrt{6}
$$

Okay, so that's the square. Then, to get the cube, I need to multiply that by $(\sqrt{2} + \sqrt{3})$:

$$
(5 + 2\sqrt{6})(\sqrt{2} + \sqrt{3})
$$

Let me expand this:

First, multiply 5 by $\sqrt{2}$: $5\sqrt{2}$

Then 5 times $\sqrt{3}$: $5\sqrt{3}$

Then $2\sqrt{6} \times \sqrt{2}$: Let me compute $\sqrt{6} \times \sqrt{2}$. That's $\sqrt{12}$, which simplifies to $2\sqrt{3}$. So 2 times that is $4\sqrt{3}$.

Then $2\sqrt{6} \times \sqrt{3}$: $\sqrt{6} \times \sqrt{3} = \sqrt{18} = 3\sqrt{2}$. So 2 times that is $6\sqrt{2}$.

So putting all those terms together:

From the first multiplication: 5√2 + 5√3

From the second multiplication: 4√3 + 6√2

Adding them all up:

For √2 terms: 5√2 + 6√2 = 11√2

For √3 terms: 5√3 + 4√3 = 9√3

So again, same result: 11√2 + 9√3. But there is no √6 term here. However, the problem says it can be written as a√2 + b√3 + c√6. So maybe there is a mistake in my calculation, or maybe c is zero?

Wait, the problem says "can be written in the form a√2 + b√3 + c√6, where a, b, and c are integers." So maybe c is zero? But why would they include that term? Maybe there's an error in my calculation?

Wait, but let me check again. Let me compute $(\sqrt{2} + \sqrt{3})^3$ numerically, and also compute my answer numerically to check.

First, let's compute the original expression numerically. Let me calculate sqrt(2) ≈ 1.4142, sqrt(3) ≈ 1.732.

So sqrt(2) + sqrt(3) ≈ 1.4142 + 1.732 ≈ 3.1462.

Then, cube that: 3.1462^3. Let me compute that:

First, 3^3 is 27. Let me compute 3.1462^3:

3.1462 * 3.1462 = ?

Let me compute 3 * 3.1462 = 9.4386, 0.1462 * 3.1462 ≈ approximately 0.1462 * 3 ≈ 0.4386, plus 0.1462 * 0.1462 ≈ ~0.0214. So total approx 0.4386 + 0.0214 ≈ 0.46. So total approx 9.4386 + 0.46 ≈ 9.8986. Then multiply by 3.1462 again:

9.8986 * 3.1462 ≈ Let's approximate:

10 * 3.1462 = 31.462

Subtract 0.1014 * 3.1462 ≈ 0.319

So approx 31.462 - 0.319 ≈ 31.143.

So approx 31.14.

Now let me compute my expression: 11√2 + 9√3.

Compute 11√2 ≈ 11 * 1.4142 ≈ 15.5562

Compute 9√3 ≈ 9 * 1.732 ≈ 15.588

Adding them together: 15.5562 + 15.588 ≈ 31.1442. Which is very close to the earlier approximation of 31.14. So that checks out. So my calculation of 11√2 + 9√3 seems correct.

But the problem says it can be written in the form a√2 + b√3 + c√6. So why is there no √6 term? Maybe the problem is written in a way that allows c to be zero? So maybe the answer is a = 11, b = 9, c = 0, so a + b + c = 20?

But why would the problem mention that form if c is zero? Maybe I made a mistake in the problem statement?

Wait, let me check again. Wait, the original problem says: "The number $(\sqrt{2}+\sqrt{3})^3$ can be written in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers. What is $a+b+c$?" So maybe they just included the √6 term as a possible term, but it's allowed for c to be zero. So maybe the answer is 11 + 9 + 0 = 20? But let me check again.

Alternatively, perhaps my expansion is missing something? Let me think. Let me try to expand $(\sqrt{2} + \sqrt{3})^3$ again step by step.

Alternatively, perhaps I should use the binomial coefficients again. Let me write the expansion again:

$(\sqrt{2} + \sqrt{3})^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(\sqrt{3}) + 3(\sqrt{2})(\sqrt{3})^2 + (\sqrt{3})^3$.

Compute each term:

First term: $(\sqrt{2})^3 = 2^{3/2} = 2 \sqrt{2}$

Second term: $3 \times (\sqrt{2})^2 \times \sqrt{3} = 3 \times 2 \times \sqrt{3} = 6 \sqrt{3}$

Third term: $3 \times \sqrt{2} \times (\sqrt{3})^2 = 3 \times \sqrt{2} \times 3 = 9 \sqrt{2}$

Fourth term: $(\sqrt{3})^3 = 3^{3/2} = 3 \sqrt{3}$

Adding all together:

First term: 2√2

Second term: 6√3

Third term: 9√2

Fourth term: 3√3

So adding the like terms:

For √2: 2√2 + 9√2 = 11√2

For √3: 6√3 + 3√3 = 9√3

No √6 terms here. So unless there is an error in my calculation, the expression is 11√2 + 9√3 + 0√6. So that would make c = 0, so a + b + c = 11 + 9 + 0 = 20. Is that the answer?

But maybe there's another approach where there is a √6 term? Let me think. Maybe I need to consider another expansion?

Wait, let me check if there's a different way to write the expression. For example, maybe combining terms?

Alternatively, could there be an error in the problem statement? Or perhaps I need to rationalize or something else? Wait, but the problem says that it can be written in that form, so maybe c is zero. Maybe the answer is 20?

Alternatively, maybe I made a mistake in my numerical calculation? Let me check with more precise numbers.

Let me calculate $(\sqrt{2} + \sqrt{3})^3$ more accurately.

First, exact value. Wait, if I have 11√2 + 9√3, then let me compute that with more precise decimal approx. Let me use more accurate values:

sqrt(2) ≈ 1.41421356

sqrt(3) ≈ 1.73205080757

So 11√2 ≈ 11 * 1.41421356 ≈ 15.55634916

9√3 ≈ 9 * 1.73205080757 ≈ 15.588457268

Adding those two gives ≈ 15.55634916 + 15.588457268 ≈ 31.14480643

Now let's compute the original expression numerically:

sqrt(2) + sqrt(3) ≈ 1.41421356 + 1.73205080757 ≈ 3.14626436757

Now, cube that:

First, square it: (3.14626436757)^2 ≈ Let me compute that:

First, 3^2 = 9, 0.14626436757^2 ≈ approx 0.0214, and cross terms:

Using (a + b)^2 = a^2 + 2ab + b^2 where a = 3, b = 0.14626436757.

So that's 9 + 2*3*0.14626436757 + (0.14626436757)^2 ≈ 9 + 0.8775862054 + 0.0214 ≈ 9.8989862054

Now multiply that by 3.14626436757:

So 9.8989862054 * 3.14626436757 ≈ Let me compute this more accurately.

Let me note that 9.8989862054 * 3 = 29.6969586162

Then, 9.8989862054 * 0.14626436757 ≈ ?

First, approximate 9.8989862054 * 0.1 = 0.98989862054

9.8989862054 * 0.04 = 0.395959448216

9.8989862054 * 0.00626436757 ≈ Let's approximate:

0.006 * 9.8989862054 ≈ 0.05939391723

Adding up: 0.98989862054 + 0.395959448216 ≈ 1.38585806876 + 0.05939391723 ≈ 1.445251986

So total approximate value is 29.6969586162 + 1.445251986 ≈ 31.1422106

Which is very close to the previous calculation of 31.14480643. The slight difference is due to rounding errors. Therefore, my expression 11√2 + 9√3 is accurate.

So, since there's no √6 term, c must be zero, so a + b + c = 11 + 9 + 0 = 20. Therefore, the answer is 20.

But wait, the problem says "can be written in the form...", so maybe there's another way to express it with a √6 term? Let me think. Suppose that maybe when expanding, there is a term with √6?

Wait, let me think again. If I consider that maybe in my expansion, I missed a cross term? Wait, but in the expansion, when we expand $(a + b)^3$, there are only four terms, as per binomial theorem. When I expanded using the square first, I only got terms with √2 and √3, no √6 terms. Wait, unless there is an error in my calculation?

Wait, let me try expanding $(\sqrt{2} + \sqrt{3})^3$ using another method. Let me consider multiplying step by step:

Let me compute $(\sqrt{2} + \sqrt{3})^3 = (\sqrt{2} + \sqrt{3}) \times (\sqrt{2} + \sqrt{3}) \times (\sqrt{2} + \sqrt{3})$

First compute the first two factors: $(\sqrt{2} + \sqrt{3})^2 = 5 + 2\sqrt{6}$ as before.

Now multiply that by $(\sqrt{2} + \sqrt{3})$:

(5 + 2√6)(√2 + √3) = 5√2 + 5√3 + 2√6√2 + 2√6√3

Compute the terms:

First term: 5√2

Second term: 5√3

Third term: 2√6 * √2 = 2√(6*2) = 2√12 = 2*2√3 = 4√3

Fourth term: 2√6 * √3 = 2√(6*3) = 2√18 = 2*3√2 = 6√2

So combining all terms:

5√2 + 5√3 + 4√3 + 6√2 = (5√2 + 6√2) + (5√3 + 4√3) = 11√2 + 9√3

Same result again. So there's no term with √6. Therefore, c is zero. Therefore, the answer is 11 + 9 + 0 = 20.

But why does the problem mention the form including c√6? Maybe it's possible that in some other expansion, there might be a √6 term? Or maybe it's just including that term as a possibility, even if c=0? Maybe the problem expects the answer to have c=0? So then the answer is 20. But maybe I need to confirm once again.

Alternatively, maybe I made a mistake in the problem statement? Wait, the problem says "the number can be written in the form a√2 + b√3 + c√6". So even if c is zero, that's allowed. So, in that case, the answer is 11 + 9 + 0 = 20. Therefore, I think that's the answer. Maybe the problem is designed to have c=0, but just include that term in the form for generality.

Alternatively, let me try to think if there is another approach where there is a √6 term. For example, maybe if we consider that when expanding, there is a term that combines to √6?

Wait, let me think. If I consider that $(\sqrt{2} + \sqrt{3})^3$ might have terms that can be combined with √6. For example, suppose we have some terms that can be expressed as a multiple of √6?

Wait, but in my previous steps, all terms were either √2 or √3. Unless I made a mistake in simplifying the terms. Let me check the third term again:

Third term when I did the expansion was 2√6 * √2. Wait, no, when I multiplied (5 + 2√6)(√2 + √3), the third term was 2√6 * √2. Let me compute that again:

√6 * √2 = √(6*2) = √12 = 2√3. So 2√6 * √2 is 2 * 2√3 = 4√3. Correct. Similarly, 2√6 * √3 is √(6*3) = √18 = 3√2, so times 2 is 6√2. Correct.

So all terms are accounted for, and there is no √6 term. Therefore, conclusion is that c=0.

Therefore, the answer is 11 + 9 + 0 = 20. Therefore, I think that is the answer.

Wait, but I'm a bit confused because the problem specifically includes the √6 term in the form. Maybe there is an alternative way to write this expression with √6? Let me think again. Suppose that I didn't use the binomial theorem, but instead tried to express it as something else. Wait, but I think that my calculations are correct.

Alternatively, maybe the problem had a typo? But assuming that the problem is correct, then the answer is 20.

Alternatively, maybe there is a miscalculation in my part? Let me check once again.

Let me try to compute $(\sqrt{2} + \sqrt{3})^3$ using another approach. Let me denote $x = \sqrt{2}$ and $y = \sqrt{3}$, so that the expression is $(x + y)^3$. Expanding this:

$x^3 + 3x^2y + 3xy^2 + y^3$. As before.

Compute each term:

$x^3 = (\sqrt{2})^3 = 2\sqrt{2}$

$3x^2y = 3*(\sqrt{2})^2*\sqrt{3} = 3*2*\sqrt{3} = 6\sqrt{3}$

$3xy^2 = 3*\sqrt{2}*(\sqrt{3})^2 = 3*\sqrt{2}*3 = 9\sqrt{2}$

$y^3 = (\sqrt{3})^3 = 3\sqrt{3}$

Adding all terms:

2√2 + 6√3 + 9√2 + 3√3 = (2 + 9)√2 + (6 + 3)√3 = 11√2 + 9√3. Same result. So there's no √6 term. So if the problem says that it can be written in that form, then c = 0. Thus, a + b + c = 11 + 9 + 0 = 20.

Therefore, I think the answer is 20. Maybe the problem includes the √6 term to trick students into thinking there's a cross term, but in reality, there isn't. So the answer is 20. Let me check online if possible? Wait, I can't check, but since my calculations are consistent across multiple methods and numerical verification aligns, I think that is correct.

Thus, the answer should be 20.

**Final Answer**
\boxed{20}

To evaluate the expression $(\sqrt{2} + \sqrt{3})^3$ and express it in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers, we begin by applying the **binomial theorem**:

$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$

Let us define:
- $a = \sqrt{2}$
- $b = \sqrt{3}$

Then the expression becomes:

$$
(\sqrt{2} + \sqrt{3})^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(\sqrt{3}) + 3(\sqrt{2})(\sqrt{3})^2 + (\sqrt{3})^3
$$

Now compute each term individually:

1. $(\sqrt{2})^3 = \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = 2\sqrt{2}$

2. $3(\sqrt{2})^2(\sqrt{3}) = 3 \cdot 2 \cdot \sqrt{3} = 6\sqrt{3}$

3. $3(\sqrt{2})(\sqrt{3})^2 = 3 \cdot \sqrt{2} \cdot 3 = 9\sqrt{2}$

4. $(\sqrt{3})^3 = \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = 3\sqrt{3}$

Now add all the terms together:

- $\sqrt{2}$ terms: $2\sqrt{2} + 9\sqrt{2} = 11\sqrt{2}$
- $\sqrt{3}$ terms: $6\sqrt{3} + 3\sqrt{3} = 9\sqrt{3}$

So, the expression simplifies to:

$$
(\sqrt{2} + \sqrt{3})^3 = 11\sqrt{2} + 9\sqrt{3}
$$

---

This is of the form:

$$
a\sqrt{2} + b\sqrt{3} + c\sqrt{6}
$$

where:
- $a = 11$
- $b = 9$
- $c = 0$

Thus, the sum $a + b + c = 11 + 9 + 0 = \boxed{20}$

---

**Final Answer:** $\boxed{20}$

Solution:
First, we calculate $(\sqrt{2} + \sqrt{3})^2$: \begin{align*}
(\sqrt{2} + \sqrt{3})^2 &= (\sqrt{2} + \sqrt{3})(\sqrt{2}+\sqrt{3})\\
&=(\sqrt{2})(\sqrt{2}) + (\sqrt{2})(\sqrt{3}) + (\sqrt{3})(\sqrt{2}) + (\sqrt{3})(\sqrt{3})\\
&= 2 + \sqrt{6} + \sqrt{6} + 3\\
&=5+2\sqrt{6}.
\end{align*} Multiplying this by $\sqrt{2} +\sqrt{3}$ gives \begin{align*}
(\sqrt{2}+ \sqrt{3})^3 &=(\sqrt{2}+\sqrt{3})^2 (\sqrt{2} +\sqrt{3})\\
&=(5+2\sqrt{6})(\sqrt{2} +\sqrt{3})\\
&= 5\sqrt{2} + 5\sqrt{3} + (2\sqrt{6})(\sqrt{2}) + (2\sqrt{6})(\sqrt{3})\\
&=5\sqrt{2} + 5\sqrt{3} + 2\sqrt{12} + 2\sqrt{18}\\
&=5\sqrt{2} + 5\sqrt{3} + 2(2\sqrt{3}) + 2(3\sqrt{2})\\
&=11\sqrt{2} + 9\sqrt{3}.
\end{align*} Therefore, we have $a+b+c = \boxed{20}$.  (Notice that $c=0;$ tricky!)

We may also expand $(\sqrt{3} + \sqrt{2})^3$ using the Binomial Theorem, giving us ${\sqrt{2}}^3 + 3{\sqrt{2}}^2\sqrt{3}+3\sqrt{2}{\sqrt{3}}^2+{\sqrt{3}}^3$. Simplifying this yields $2\sqrt{2}+6\sqrt{3}+9\sqrt{2}+3\sqrt{3} = 11\sqrt{2}+9\sqrt{3}$, and once again $a + b + c = \boxed{20}$.