<Question>
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 $a+b+c$?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how to expand (√2 + √3)^3 and express it in the form a√2 + b√3 + c√6, where a, b, and c are integers. Then, I have to find the sum a + b + c. Hmm, let's start by recalling how to expand a binomial raised to a power. The binomial theorem might come in handy here.

<prompt_1.1> Expand (√2 + √3)^3 using the binomial theorem as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The binomial theorem states that (a + b)^n = Σ (from k=0 to n) [C(n, k) * a^(n-k) * b^k], where C(n, k) is the combination of n things taken k at a time. In this case, a is √2, b is √3, and n is 3. So maybe I can apply the binomial theorem here.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me write that out step by step. 

First, expand (√2 + √3)^3 using the binomial expansion:

= C(3,0)*(√2)^3 + C(3,1)*(√2)^2*(√3) + C(3,2)*(√2)*(√3)^2 + C(3,3)*(√3)^3

Now, let's compute each term individually.

<prompt_2.1> Compute each term of the expanded expression as per the above plan. </prompt_2.1>
-----

Alternatively, maybe I can verify by multiplying (√2 + √3)^3 step by step. Let's try that.

First, compute (√2 + √3)^2:

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

Then, multiply this by (√2 + √3) to get the cube:

(5 + 2√6)(√2 + √3)

Let's distribute each term:

= 5*√2 + 5*√3 + 2√6*√2 + 2√6*√3

Simplify each term:

<prompt_2.2> Verify the result by multiplying (√2 + √3)^2 by (√2 + √3) step by step as per the above plan. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
1. First term: C(3,0)*(√2)^3 = 1*(√2)^3. Since C(3,0) is 1. Now, (√2)^3 is (2)^(3/2) which is 2*√2. So this term is 2√2.

2. Second term: C(3,1)*(√2)^2*(√3). C(3,1) is 3. (√2)^2 is 2, so this term becomes 3*2*√3 = 6√3.

3. Third term: C(3,2)*(√2)*(√3)^2. C(3,2) is 3. (√3)^2 is 3, so this term is 3*√2*3 = 9√2.

4. Fourth term: C(3,3)*(√3)^3. C(3,3) is 1. (√3)^3 is 3√3. So this term is 1*3√3 = 3√3.
</execution_2.1>

<execution_2.2>
- 5√2 remains as is.
- 5√3 remains as is.
- 2√6*√2 = 2√(6*2) = 2√12 = 2*2√3 = 4√3
- 2√6*√3 = 2√(6*3) = 2√18 = 2*3√2 = 6√2

Now, combine all the terms:

5√2 + 5√3 + 4√3 + 6√2

Combine like terms:

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

Wait, that's the same result as before. So, according to both methods, the expansion is 11√2 + 9√3, which means there's no √6 term. Therefore, c = 0. Hence, a + b + c = 11 + 9 + 0 = 20.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now, let's add up all the terms:

<prompt_3.1> Add up all the computed terms to express (√2 + √3)^3 in the form a√2 + b√3 + c√6 as per the above plan. </prompt_3.1>
-----

Based on execution_2.2:
But wait, let me double-check my calculations to make sure I didn't make any mistakes. Let's verify each step again.

<prompt_3.2> Double-check the calculations from both methods to ensure the result is correct as per the above plan. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
First term: 2√2

Second term: 6√3

Third term: 9√2

Fourth term: 3√3

Adding the √2 terms: 2√2 + 9√2 = 11√2

Adding the √3 terms: 6√3 + 3√3 = 9√3

The third term didn't have a √6 component, right? Let me check each term again.

Wait, hold on. Let me verify each term's simplification to ensure I didn't make a mistake.

First term: (√2)^3 = (√2)^2 * √2 = 2 * √2 = 2√2. Correct.

Second term: (√2)^2 = 2, so 3 * 2 * √3 = 6√3. Correct.

Third term: (√3)^2 = 3, so 3 * √2 * 3 = 9√2. Correct.

Fourth term: (√3)^3 = (√3)^2 * √3 = 3 * √3 = 3√3. So 1 * 3√3 = 3√3. Correct.

So adding the √2 terms: 2√2 + 9√2 = 11√2. Correct.

Adding the √3 terms: 6√3 + 3√3 = 9√3. Correct.

But wait, the problem statement says the form is a√2 + b√3 + c√6. In my result, there is no √6 term. That would mean c is zero.
</execution_3.1>

<execution_3.2>
First method: Binomial expansion.

First term: (√2)^3 = (√2)^2 * √2 = 2 * √2 = 2√2. Correct.

Second term: 3*(√2)^2*(√3) = 3*2*√3 = 6√3. Correct.

Third term: 3*(√2)*(√3)^2 = 3*√2*3 = 9√2. Correct.

Fourth term: (√3)^3 = (√3)^2*√3 = 3*√3 = 3√3. Correct.

Adding them up: 2√2 + 6√3 + 9√2 + 3√3 = (2 + 9)√2 + (6 + 3)√3 = 11√2 + 9√3. That seems right.

Second method: Multiplying step by step.

(√2 + √3)^2 = 5 + 2√6. Correct, since (a + b)^2 = a^2 + 2ab + b^2, which gives 2 + 2√6 + 3 = 5 + 2√6.

Then multiplying by (√2 + √3):

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

Calculating each term:

5√2 is correct.

5√3 is correct.

2√6*√2: √6*√2 = √(6*2) = √12 = 2√3, so 2*2√3 = 4√3. Correct.

2√6*√3: √6*√3 = √(6*3) = √18 = 3√2, so 2*3√2 = 6√2. Correct.

Adding up: 5√2 + 6√2 = 11√2; 5√3 + 4√3 = 9√3. Correct.

Therefore, both methods give the same result. So, the coefficients are a = 11, b = 9, c = 0. Therefore, a + b + c = 11 + 9 + 0 = 20.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
But the problem states that it should be in the form a√2 + b√3 + c√6. So, does that mean c is zero? Let me check if I missed any √6 terms in the expansion.

Looking back at the terms:

<prompt_4.1> Re-evaluate the expansion to confirm the absence of √6 terms as per the above plan. </prompt_4.1>
-----

Based on execution_3.2:
But wait, just to be thorough, let me compute the numerical values to confirm both the original expression and the expanded form give the same result.

<prompt_4.2> Compute numerical values to confirm both the original expression and the expanded form as per the above plan. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
1. 2√2 (no √3)
2. 6√3 (no √2)
3. 9√2 (no √3)
4. 3√3 (no √2)

Wait, but when we multiplied the terms, did any cross terms involving √6 appear? Let me check again.

Wait, when we have terms like (√2)^2*(√3) which is 2*√3, that's just √3. Similarly, (√2)*(√3)^2 is √2*3, which is √2. The only terms that would involve √6 are if we had a product of √2 and √3, but in the expansion, the cross terms when multiplying (√2 + √3)(√2 + √3)(√2 + √3) might result in √6. Wait, but in the binomial expansion, we already accounted for all terms. Let me check each term again.

Wait, perhaps I made a mistake in the binomial expansion. Let me re-examine that.

Wait, when expanding (√2 + √3)^3, the terms are:

C(3,0)*(√2)^3*(√3)^0 = 1*2√2*1 = 2√2

C(3,1)*(√2)^2*(√3)^1 = 3*2*√3 = 6√3

C(3,2)*(√2)^1*(√3)^2 = 3*√2*3 = 9√2

C(3,3)*(√2)^0*(√3)^3 = 1*1*3√3 = 3√3

So adding those up: 2√2 + 6√3 + 9√2 + 3√3 = (2 + 9)√2 + (6 + 3)√3 = 11√2 + 9√3. So indeed, there is no √6 term. Therefore, c = 0. Hence, a + b + c = 11 + 9 + 0 = 20.
</execution_4.1>

<execution_4.2>
Let's compute (√2 + √3)^3 numerically.

First, approximate √2 ≈ 1.4142, √3 ≈ 1.732.

Compute √2 + √3 ≈ 1.4142 + 1.732 ≈ 3.1462

Cube that: (3.1462)^3 ≈ 3.1462 * 3.1462 * 3.1462

First, compute 3.1462 * 3.1462 ≈ 9.879 (since π is approximately 3.1416, but this is a rough estimate). Let's say approximately 9.879.

Then multiply by 3.1462: 9.879 * 3.1462 ≈ 31.07 (since 10 * 3.1462 = 31.462, subtract 0.121 * 3.1462 ≈ 0.382, so 31.462 - 0.382 ≈ 31.08).

Now, compute 11√2 + 9√3 ≈ 11*1.4142 + 9*1.732 ≈ 15.5562 + 15.588 ≈ 31.144.

The original expression's cube was approximately 31.08, and the expanded form is approximately 31.14. The slight difference is due to rounding errors in the decimal approximations. Therefore, this confirms that the expansion is correct.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.1, execution_4.2:
Therefore, the coefficients are a = 11, b = 9, c = 0, so a + b + c = 20. 

But just to be absolutely thorough, let me check if there's any possible mistake in the binomial expansion. Let's re-examine each term:

<prompt_5.1> Perform a final verification of the binomial expansion to ensure no mistakes were made as per the above plan. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
1. First term: C(3,0)*(√2)^3 = 1*(√2)^3 = (√2)^2*√2 = 2*√2 = 2√2. Correct.

2. Second term: C(3,1)*(√2)^2*(√3) = 3*(2)*(√3) = 6√3. Correct.

3. Third term: C(3,2)*(√2)*(√3)^2 = 3*(√2)*(3) = 9√2. Correct.

4. Fourth term: C(3,3)*(√3)^3 = 1*(3√3) = 3√3. Correct.

Adding up all the √2 terms: 2√2 + 9√2 = 11√2. Correct.

Adding up all the √3 terms: 6√3 + 3√3 = 9√3. Correct.

So, the expansion is indeed 11√2 + 9√3. Therefore, the coefficients a, b, c are 11, 9, 0 respectively. Hence, a + b + c = 11 + 9 + 0 = 20.
</execution_5.1>
</Execution_5>

<Final_answer>
a + b + c = 11 + 9 + 0 = \boxed{20}.

\]
</Final_answer>
</think>