The coefficient of `x^(4)` in the expansion of `(1+x+x^(2)+x^(3))^(11)` is where `({:(n),(r):})`= `nC_(r)`

Options

A. `({:(11),(4):})`
B. `({:(11),(4):})+({:(11),(2):})`
C. `({:(11),(4):})+({:(11),(2):})+({:(11),(4):}).({:(11),(2):})`
D. `({:(11),(4):})+({:(11),(2):})+({:(11),(1):})({:(11),(2):})`

Explanation

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 rewritten as a geometric series:
\[ 1 + x + x^2 + x^3 = \frac{1 - x^4}{1 - x} \]
This means we can express our original problem as:
\[ (1 + x + x^2 + x^3)^{11} = \left( \frac{1 - x^4}{1 - x} \right)^{11} \]

### Step 2: Expand the expression
Using the binomial theorem, we can expand this:
\[ (1 - x^4)^{11} \cdot (1 - x)^{-11} \]
We will need to find the coefficient of \( x^4 \) in this expansion.

### Step 3: Find the coefficient of \( x^4 \) in \( (1 - x^4)^{11} \)
Using the binomial expansion:
\[ (1 - x^4)^{11} = \sum_{k=0}^{11} \binom{11}{k} (-1)^k x^{4k} \]
We need the term where \( 4k = 4 \), which gives \( k = 1 \). The coefficient is:
\[ \binom{11}{1} (-1)^1 = -11 \]

### Step 4: Find the coefficient of \( x^0 \) in \( (1 - x)^{-11} \)
Using the binomial series expansion:
\[ (1 - x)^{-11} = \sum_{m=0}^{\infty} \binom{m + 10}{10} x^m \]
The coefficient of \( x^0 \) is:
\[ \binom{10}{10} = 1 \]

### Step 5: Combine the results
The coefficient of \( x^4 \) in the overall expansion is given by:
\[ \text{Coefficient of } x^4 = (-11) \cdot 1 = -11 \]

### Step 6: Find the coefficient of \( x^4 \) in \( (1 - x)^{-11} \)
We also need to consider the terms from \( (1 - x)^{-11} \) that contribute to \( x^4 \):
\[ \text{Coefficient of } x^4 = \binom{10 + 4}{4} = \binom{14}{4} = 1001 \]

### Step 7: Combine the contributions
Now, we combine the contributions:
\[ \text{Total coefficient of } x^4 = -11 + 1001 = 990 \]

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