Find the coefficient of
`x^4` in the expansion of `(1+x+x^2 +x^3 )^11`

Answer

To find the coefficient of \( x^4 \) in the expansion of \( (1 + x + x^2 + x^3)^{11} \), we can follow these steps:

### Step 1: Rewrite the Expression
The expression \( 1 + x + x^2 + x^3 \) can be factored. Notice that:
\[ 1 + x + x^2 + x^3 = (1 + x)(1 + x^2) \]
Thus, we can rewrite the original expression as:
\[ (1 + x + x^2 + x^3)^{11} = [(1 + x)(1 + x^2)]^{11} \]

### Step 2: Apply the Binomial Theorem
Using the binomial theorem, we can expand \( [(1 + x)(1 + x^2)]^{11} \):
\[ [(1 + x)(1 + x^2)]^{11} = (1 + x)^{11} (1 + x^2)^{11} \]

### Step 3: Expand Each Factor
Now we need to expand both \( (1 + x)^{11} \) and \( (1 + x^2)^{11} \).

1. **Expansion of \( (1 + x)^{11} \)**:
The general term in this expansion is given by:
\[ T_k = \binom{11}{k} x^k \]
for \( k = 0, 1, 2, \ldots, 11 \).

2. **Expansion of \( (1 + x^2)^{11} \)**:
The general term in this expansion is:
\[ S_m = \binom{11}{m} (x^2)^m = \binom{11}{m} x^{2m} \]
for \( m = 0, 1, 2, \ldots, 11 \).

### Step 4: Combine the Expansions
To find the coefficient of \( x^4 \) in the product, we need to consider all pairs \( (k, m) \) such that:
\[ k + 2m = 4 \]

### Step 5: Find Valid Pairs
We can find valid pairs \( (k, m) \):
- If \( m = 0 \), then \( k + 0 = 4 \) → \( k = 4 \)
- If \( m = 1 \), then \( k + 2 = 4 \) → \( k = 2 \)
- If \( m = 2 \), then \( k + 4 = 4 \) → \( k = 0 \)

Thus, the valid pairs are:
1. \( (k, m) = (4, 0) \)
2. \( (k, m) = (2, 1) \)
3. \( (k, m) = (0, 2) \)

### Step 6: Calculate Coefficients
Now we calculate the contributions from each pair:
1. For \( (4, 0) \):
\[ \text{Coefficient} = \binom{11}{4} \cdot \binom{11}{0} = 330 \cdot 1 = 330 \]

2. For \( (2, 1) \):
\[ \text{Coefficient} = \binom{11}{2} \cdot \binom{11}{1} = 55 \cdot 11 = 605 \]

3. For \( (0, 2) \):
\[ \text{Coefficient} = \binom{11}{0} \cdot \binom{11}{2} = 1 \cdot 55 = 55 \]

### Step 7: Sum the Coefficients
Now, we sum all the contributions:
\[ \text{Total Coefficient of } x^4 = 330 + 605 + 55 = 990 \]

### Final Answer
The coefficient of \( x^4 \) in the expansion of \( (1 + x + x^2 + x^3)^{11} \) is **990**.