The coefficients of `x^(13)` in the expansion of
` (1 - x)^(5) (1 + x + x^(2) + x^(3) )^(4)` , is

Options

A. 4
B. -4
C. 9
D. none of these

Explanation

To find the coefficient of \( x^{13} \) in the expansion of \( (1 - x)^{5} (1 + x + x^{2} + x^{3})^{4} \), we can follow these steps:

### Step 1: Simplify the expression
First, we simplify \( (1 + x + x^{2} + x^{3})^{4} \). This can be rewritten as:
\[ (1 - x^4)^{4} \cdot (1 + x)^{4} \]
This is because \( 1 + x + x^{2} + x^{3} = \frac{1 - x^{4}}{1 - x} \).

### Step 2: Combine the expressions
Now, we can rewrite the entire expression as:
\[ (1 - x)^{5} \cdot (1 - x^{4})^{4} \cdot (1 + x)^{4} \]

### Step 3: Expand the components
Using the Binomial Theorem, we can expand each component:
1. \( (1 - x)^{5} = \sum_{k=0}^{5} \binom{5}{k} (-1)^{k} x^{k} \)
2. \( (1 - x^{4})^{4} = \sum_{m=0}^{4} \binom{4}{m} (-1)^{m} x^{4m} \)
3. \( (1 + x)^{4} = \sum_{n=0}^{4} \binom{4}{n} x^{n} \)

### Step 4: Find the coefficient of \( x^{13} \)
To find the coefficient of \( x^{13} \), we need to consider the combinations of terms from each expansion that will yield \( x^{13} \).

From \( (1 - x)^{5} \), we can take terms \( x^{k} \) where \( k \) can be from 0 to 5.
From \( (1 - x^{4})^{4} \), we can take terms \( x^{4m} \) where \( m \) can be from 0 to 4.
From \( (1 + x)^{4} \), we can take terms \( x^{n} \) where \( n \) can be from 0 to 4.

We need to satisfy the equation:
\[ k + 4m + n = 13 \]

### Step 5: Calculate the coefficients
We will consider different values of \( m \) (from \( 0 \) to \( 3 \)), as \( 4m \) can take values \( 0, 4, 8, 12 \).

1. **For \( m = 0 \)**:
\[ k + n = 13 \quad (k \leq 5, n \leq 4) \Rightarrow \text{No valid combinations} \]

2. **For \( m = 1 \)**:
\[ k + n = 9 \quad (k \leq 5, n \leq 4) \Rightarrow \text{No valid combinations} \]

3. **For \( m = 2 \)**:
\[ k + n = 5 \quad (k \leq 5, n \leq 4) \]
Valid combinations:
- \( k = 5, n = 0 \): Coefficient = \( \binom{5}{5} \cdot \binom{4}{2} \cdot \binom{4}{0} = 1 \cdot 6 \cdot 1 = 6 \)
- \( k = 4, n = 1 \): Coefficient = \( \binom{5}{4} \cdot \binom{4}{2} \cdot \binom{4}{1} = 5 \cdot 6 \cdot 4 = 120 \)
- \( k = 3, n = 2 \): Coefficient = \( \binom{5}{3} \cdot \binom{4}{2} \cdot \binom{4}{2} = 10 \cdot 6 \cdot 6 = 360 \)
- \( k = 2, n = 3 \): Coefficient = \( \binom{5}{2} \cdot \binom{4}{2} \cdot \binom{4}{3} = 10 \cdot 6 \cdot 4 = 240 \)
- \( k = 1, n = 4 \): Coefficient = \( \binom{5}{1} \cdot \binom{4}{2} \cdot \binom{4}{4} = 5 \cdot 6 \cdot 1 = 30 \)

Total for \( m = 2 \): \( 6 + 120 + 360 + 240 + 30 = 756 \)

4. **For \( m = 3 \)**:
\[ k + n = 1 \quad (k \leq 5, n \leq 4) \]
Valid combinations:
- \( k = 1, n = 0 \): Coefficient = \( \binom{5}{1} \cdot \binom{4}{3} \cdot \binom{4}{0} = 5 \cdot 4 \cdot 1 = 20 \)
- \( k = 0, n = 1 \): Coefficient = \( \binom{5}{0} \cdot \binom{4}{3} \cdot \binom{4}{1} = 1 \cdot 4 \cdot 4 = 16 \)

Total for \( m = 3 \): \( 20 + 16 = 36 \)

### Final Calculation
Now, we sum the coefficients from all valid combinations:
\[ \text{Total Coefficient of } x^{13} = 756 + 36 = 792 \]

### Conclusion
The coefficient of \( x^{13} \) in the expansion of \( (1 - x)^{5} (1 + x + x^{2} + x^{3})^{4} \) is **792**.