The coefficient of `x^4` in the expansion of `(1+x+x^2+x^3)^n` is

Options

A. `""^(n)C_(4)`
B. `""^(n)C_(4)+""^(n)C_(2)`
C. `""^(n)C_(4)+""^(n)C_(1)+""^(n)C_(4).""^(n)C_(2)`
D. `""^(n)C_(4)+""^(n)C_(2)+""^(n)C_(1).""^(n)C_(2)`

Answer

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

### Step 1: Rewrite the expression
The expression \( 1 + x + x^2 + x^3 \) can be rewritten using the formula for the sum of a geometric series. We can factor it as follows:
\[ 1 + x + x^2 + x^3 = \frac{1 - x^4}{1 - x} \]
This means that:
\[ (1 + x + x^2 + x^3)^n = \left(\frac{1 - x^4}{1 - x}\right)^n \]

### Step 2: Expand the expression
Using the binomial theorem, we can expand \( (1 - x^4)^n \) and \( (1 - x)^{-n} \):
\[ (1 - x^4)^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k x^{4k} \]
\[ (1 - x)^{-n} = \sum_{m=0}^{\infty} \binom{n + m - 1}{m} x^m \]

### Step 3: Find the coefficient of \( x^4 \)
To find the coefficient of \( x^4 \) in the product of these two expansions, we need to consider the combinations of terms that yield \( x^4 \):
1. From \( (1 - x^4)^n \), we can take \( x^0 \) (the constant term) and \( x^4 \) (the term with \( k=1 \)).
2. From \( (1 - x)^{-n} \), we can take \( x^4 \) (the term with \( m=4 \)).

Thus, the coefficient of \( x^4 \) can be calculated as follows:
\[ \text{Coefficient of } x^4 = \text{Coefficient of } x^4 \text{ from } (1 - x^4)^n \cdot \text{Coefficient of } x^0 \text{ from } (1 - x)^{-n} + \text{Coefficient of } x^0 \text{ from } (1 - x^4)^n \cdot \text{Coefficient of } x^4 \text{ from } (1 - x)^{-n} \]

Calculating these:
- Coefficient of \( x^4 \) from \( (1 - x^4)^n \) is \( \binom{n}{1} (-1)^1 = -n \).
- Coefficient of \( x^0 \) from \( (1 - x)^{-n} \) is \( \binom{n + 0 - 1}{0} = 1 \).
- Coefficient of \( x^0 \) from \( (1 - x^4)^n \) is \( \binom{n}{0} = 1 \).
- Coefficient of \( x^4 \) from \( (1 - x)^{-n} \) is \( \binom{n + 4 - 1}{4} = \binom{n + 3}{4} \).

Putting it all together:
\[ \text{Coefficient of } x^4 = -n \cdot 1 + 1 \cdot \binom{n + 3}{4} = \binom{n + 3}{4} - n \]

### Final Answer
The coefficient of \( x^4 \) in the expansion of \( (1 + x + x^2 + x^3)^n \) is:
\[ \binom{n + 3}{4} - n \]