The coefficient of `x^11` in the expansion of `(1- 2x + 3x^2) (1 + x)^11 ` is

Options

A. 164
B. 144
C. 116
D. 261

Answer

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

### Step 1: Expand \( (1 + x)^{11} \)
Using the Binomial Theorem, we can expand \( (1 + x)^{11} \):
\[ (1 + x)^{11} = \sum_{r=0}^{11} \binom{11}{r} x^r \]
This means the coefficient of \( x^r \) in this expansion is \( \binom{11}{r} \).

### Step 2: Identify the terms contributing to \( x^{11} \)
We need to find the combinations of terms from \( (1 - 2x + 3x^2) \) and \( (1 + x)^{11} \) that will give us \( x^{11} \).

1. **Choosing \( 1 \) from \( (1 - 2x + 3x^2) \)**:
- We need \( x^{11} \) from \( (1 + x)^{11} \).
- Coefficient: \( \binom{11}{11} = 1 \)

2. **Choosing \( -2x \)**:
- We need \( x^{10} \) from \( (1 + x)^{11} \).
- Coefficient: \( -2 \cdot \binom{11}{10} = -2 \cdot 11 = -22 \)

3. **Choosing \( 3x^2 \)**:
- We need \( x^9 \) from \( (1 + x)^{11} \).
- Coefficient: \( 3 \cdot \binom{11}{9} = 3 \cdot \binom{11}{2} = 3 \cdot \frac{11 \cdot 10}{2 \cdot 1} = 3 \cdot 55 = 165 \)

### Step 3: Combine the coefficients
Now, we sum the contributions from each case:
\[ \text{Total coefficient} = 1 + (-22) + 165 = 1 - 22 + 165 = 144 \]

### Final Answer
The coefficient of \( x^{11} \) in the expansion of \( (1 - 2x + 3x^2)(1 + x)^{11} \) is \( \boxed{144} \).