The term independent of x in the expansion of `(1 - x)^(2) (x + (1)/(x))^10`, is

Options

A. `""^(11)C_(5)`
B. `""^(10)C_(5)`
C. `""^(10)C_(4)`
D. none of these

Answer

To find the term independent of \( x \) in the expansion of \( (1 - x)^2 \left( x + \frac{1}{x} \right)^{10} \), we will follow these steps:

### Step 1: Expand \( (1 - x)^2 \)
Using the binomial theorem, we can expand \( (1 - x)^2 \):
\[ (1 - x)^2 = 1 - 2x + x^2 \]

### Step 2: Expand \( \left( x + \frac{1}{x} \right)^{10} \)
Using the binomial theorem again, we can expand \( \left( x + \frac{1}{x} \right)^{10} \):
\[ \left( x + \frac{1}{x} \right)^{10} = \sum_{r=0}^{10} \binom{10}{r} x^r \left( \frac{1}{x} \right)^{10-r} = \sum_{r=0}^{10} \binom{10}{r} x^{2r - 10} \]

### Step 3: Combine the expansions
Now we need to combine the expansions of \( (1 - 2x + x^2) \) and \( \left( x + \frac{1}{x} \right)^{10} \):
\[ (1 - 2x + x^2) \left( \sum_{r=0}^{10} \binom{10}{r} x^{2r - 10} \right) \]

### Step 4: Identify the term independent of \( x \)
We need to find the terms where the power of \( x \) is zero (i.e., \( x^0 \)) in the combined expansion. We will consider each term from \( (1 - 2x + x^2) \):

1. **From \( 1 \)**:
- We need \( 2r - 10 = 0 \) which gives \( r = 5 \).
- The coefficient is \( \binom{10}{5} \).

2. **From \( -2x \)**:
- We need \( 2r - 10 - 1 = 0 \) which gives \( r = \frac{11}{2} \) (not possible).

3. **From \( x^2 \)**:
- We need \( 2r - 10 + 2 = 0 \) which gives \( r = 4 \).
- The coefficient is \( \binom{10}{4} \).

### Step 5: Calculate the coefficients
Now we can compute the coefficients:
- For \( r = 5 \):
\[ \text{Coefficient} = \binom{10}{5} = 252 \]

- For \( r = 4 \):
\[ \text{Coefficient} = \binom{10}{4} = 210 \]

### Step 6: Combine the coefficients
Now, we combine the contributions:
\[ \text{Independent term} = \binom{10}{5} + (-2) \cdot 0 + \binom{10}{4} = 252 + 0 + 210 = 462 \]

### Final Answer
The term independent of \( x \) in the expansion is \( 462 \).

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