The coefficient of `x^(6)` in the expansion of `(1+x+x^(2)+x^(3))(1-x)^(6)` is

Options

A. `-10`
B. 10
C. 9
D. `-9

Answer

To find the coefficient of \( x^6 \) in the expansion of \( (1 + x + x^2 + x^3)(1 - x)^6 \), we will follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ (1 + x + x^2 + x^3)(1 - x)^6 \]
We can factor out \( (1 + x + x^2 + x^3) \) as:
\[ (1 + x)(1 + x^2)(1 - x)^6 \]

2. **Expand \( (1 - x)^6 \)**:
Using the Binomial Theorem, we can expand \( (1 - x)^6 \):
\[ (1 - x)^6 = \sum_{k=0}^{6} \binom{6}{k} (-1)^k x^k \]

3. **Combine Terms**:
Now, we need to consider the product:
\[ (1 + x)(1 + x^2) \cdot \sum_{k=0}^{6} \binom{6}{k} (-1)^k x^k \]
First, we expand \( (1 + x)(1 + x^2) \):
\[ (1 + x)(1 + x^2) = 1 + x + x^2 + x^3 \]

4. **Identify Terms for \( x^6 \)**:
We need to find the coefficient of \( x^6 \) in the product. This can be achieved by considering the contributions from \( x^0, x^1, x^2, x^3 \) from \( (1 + x + x^2 + x^3) \) and the corresponding terms from \( (1 - x)^6 \).

- From \( 1 \): We need \( x^6 \) from \( (1 - x)^6 \) which is \( \binom{6}{6}(-1)^6 = 1 \).
- From \( x \): We need \( x^5 \) from \( (1 - x)^6 \) which is \( \binom{6}{5}(-1)^5 = -6 \).
- From \( x^2 \): We need \( x^4 \) from \( (1 - x)^6 \) which is \( \binom{6}{4}(-1)^4 = 15 \).
- From \( x^3 \): We need \( x^3 \) from \( (1 - x)^6 \) which is \( \binom{6}{3}(-1)^3 = -20 \).

5. **Combine Coefficients**:
Now, we sum these contributions:
\[ 1 - 6 + 15 - 20 = 1 - 6 + 15 - 20 = -10 \]

Thus, the coefficient of \( x^6 \) in the expansion of \( (1 + x + x^2 + x^3)(1 - x)^6 \) is \( -10 \).

### Final Answer:
The coefficient of \( x^6 \) is \( -10 \).