Coefficient of `x^(7)` in the expansion of `(1+x+x^(2))^(4)` is ___________

Answer

To find the coefficient of \( x^7 \) in the expansion of \( (1 + x + x^2)^4 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1 + x + x^2)^4 \]
This can be treated as a polynomial where we need to find the coefficient of \( x^7 \).

### Step 2: Use the multinomial theorem
The multinomial expansion states that:
\[ (a_1 + a_2 + a_3)^n = \sum \frac{n!}{k_1! k_2! k_3!} a_1^{k_1} a_2^{k_2} a_3^{k_3} \]
where \( k_1 + k_2 + k_3 = n \). In our case, \( a_1 = 1 \), \( a_2 = x \), \( a_3 = x^2 \), and \( n = 4 \).

### Step 3: Identify the terms contributing to \( x^7 \)
We need to find combinations of \( k_1, k_2, k_3 \) such that:
\[ k_2 + 2k_3 = 7 \]
and
\[ k_1 + k_2 + k_3 = 4 \]

### Step 4: Solve the equations
From the equations, we can express \( k_1 \) in terms of \( k_2 \) and \( k_3 \):
\[ k_1 = 4 - k_2 - k_3 \]
Substituting \( k_1 \) into the second equation gives:
\[ k_2 + 2k_3 = 7 \implies k_2 = 7 - 2k_3 \]
Substituting this into the equation for \( k_1 \):
\[ k_1 = 4 - (7 - 2k_3) - k_3 = 4 - 7 + 2k_3 - k_3 = 2k_3 - 3 \]

### Step 5: Find valid combinations
Now we need \( k_1, k_2, k_3 \) to be non-negative integers. Thus:
1. \( k_3 \geq 0 \)
2. \( 2k_3 - 3 \geq 0 \implies k_3 \geq 1.5 \) (so \( k_3 \geq 2 \))
3. \( 7 - 2k_3 \geq 0 \implies k_3 \leq 3.5 \) (so \( k_3 \leq 3 \))

Thus, \( k_3 \) can take values \( 2 \) or \( 3 \).

### Step 6: Calculate coefficients for valid \( k_3 \)
1. **For \( k_3 = 2 \)**:
- \( k_2 = 7 - 2 \cdot 2 = 3 \)
- \( k_1 = 4 - 3 - 2 = -1 \) (not valid)

2. **For \( k_3 = 3 \)**:
- \( k_2 = 7 - 2 \cdot 3 = 1 \)
- \( k_1 = 4 - 1 - 3 = 0 \) (valid)

### Step 7: Calculate the coefficient
Using the multinomial coefficient:
\[ \text{Coefficient} = \frac{4!}{0! \cdot 1! \cdot 3!} = \frac{24}{1 \cdot 6} = 4 \]

### Final Answer
Thus, the coefficient of \( x^7 \) in the expansion of \( (1 + x + x^2)^4 \) is:
\[ \boxed{4} \]