To find the coefficient of \( x^6 \) in the expansion of \( (3 + 2x + x^2)^6 \), we can use the multinomial expansion theorem. Let's break down the steps:
### Step 1: Understanding the Multinomial Expansion
The expression \( (a + b + c)^n \) can be expanded using the multinomial theorem, which states:
\[
(a + b + c)^n = \sum_{i+j+k=n} \frac{n!}{i!j!k!} a^i b^j c^k
\]
where \( i, j, k \) are non-negative integers that sum to \( n \).
### Step 2: Identify the Terms
In our case, we have:
- \( a = 3 \)
- \( b = 2x \)
- \( c = x^2 \)
- \( n = 6 \)
We need to find the coefficient of \( x^6 \) in the expansion of \( (3 + 2x + x^2)^6 \).
### Step 3: Set Up the Equation
We need to find combinations of \( i, j, k \) such that:
\[
j + 2k = 6 \quad \text{and} \quad i + j + k = 6
\]
From the second equation, we can express \( i \) as:
\[
i = 6 - j - k
\]
### Step 4: Solve for \( j \) and \( k \)
We can substitute \( i \) into the first equation:
\[
j + 2k = 6
\]
This gives us two equations:
1. \( j + k \leq 6 \)
2. \( j + 2k = 6 \)
From \( j + 2k = 6 \), we can express \( j \) in terms of \( k \):
\[
j = 6 - 2k
\]
### Step 5: Find Valid Combinations
Now we can find valid integer values for \( k \):
- If \( k = 0 \), then \( j = 6 \) and \( i = 0 \)
- If \( k = 1 \), then \( j = 4 \) and \( i = 1 \)
- If \( k = 2 \), then \( j = 2 \) and \( i = 2 \)
- If \( k = 3 \), then \( j = 0 \) and \( i = 3 \)
### Step 6: Calculate Coefficients
Now we can calculate the coefficients for each valid combination:
1. For \( (i, j, k) = (0, 6, 0) \):
\[
\text{Coefficient} = \frac{6!}{0!6!0!} \cdot 3^0 \cdot (2x)^6 \cdot (x^2)^0 = 1 \cdot 64 = 64
\]
2. For \( (i, j, k) = (1, 4, 1) \):
\[
\text{Coefficient} = \frac{6!}{1!4!1!} \cdot 3^1 \cdot (2x)^4 \cdot (x^2)^1 = 30 \cdot 3 \cdot 16 \cdot 1 = 1440
\]
3. For \( (i, j, k) = (2, 2, 2) \):
\[
\text{Coefficient} = \frac{6!}{2!2!2!} \cdot 3^2 \cdot (2x)^2 \cdot (x^2)^2 = 90 \cdot 9 \cdot 4 \cdot 1 = 3240
\]
4. For \( (i, j, k) = (3, 0, 3) \):
\[
\text{Coefficient} = \frac{6!}{3!0!3!} \cdot 3^3 \cdot (2x)^0 \cdot (x^2)^3 = 20 \cdot 27 \cdot 1 = 540
\]
### Step 7: Sum the Coefficients
Now, we sum these coefficients to find the total coefficient of \( x^6 \):
\[
64 + 1440 + 3240 + 540 = 5284
\]
### Final Answer
Thus, the coefficient of \( x^6 \) in the expansion of \( (3 + 2x + x^2)^6 \) is \( \boxed{5284} \).
