The middle term in the expansion of `(1 - (1)/(x))^(n) (1 - x)^(n)` is

Options

A. `""^(2n)C_(n)`
B. `-""^(2n)C_(n)`
C. `- ""^(2n)C_(n-1)`
D. none of these

Answer

To find the middle term in the expansion of \((1 - \frac{1}{x})^n (1 - x)^n\), we can follow these steps:

### Step 1: Understand the Expression
The expression is a product of two binomial expansions: \((1 - \frac{1}{x})^n\) and \((1 - x)^n\). We will first expand each of these separately.

### Step 2: Expand Each Binomial
1. **Expand \((1 - \frac{1}{x})^n\)** using the Binomial Theorem:
\[ (1 - \frac{1}{x})^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \left(\frac{1}{x}\right)^k = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \frac{1}{x^k} \]

2. **Expand \((1 - x)^n\)** using the Binomial Theorem:
\[ (1 - x)^n = \sum_{j=0}^{n} \binom{n}{j} (-1)^j x^j \]

### Step 3: Combine the Expansions
The product of these two expansions can be expressed as:
\[ (1 - \frac{1}{x})^n (1 - x)^n = \left(\sum_{k=0}^{n} \binom{n}{k} (-1)^k \frac{1}{x^k}\right) \left(\sum_{j=0}^{n} \binom{n}{j} (-1)^j x^j\right) \]

### Step 4: Determine the Middle Term
To find the middle term, we need to consider the total number of terms in the expansion. The total degree of the combined expansion is \(n + n = 2n\).

- If \(2n\) is even, the middle term is given by:
\[ T_{\frac{2n}{2} + 1} = T_{n + 1} \]
- If \(2n\) is odd, the middle terms are \(T_{\frac{2n + 1}{2}}\) and \(T_{\frac{2n + 3}{2}}\).

Since \(2n\) is always even for integer \(n\), we will use the formula for the middle term.

### Step 5: Find the Specific Middle Term
The middle term \(T_{n + 1}\) corresponds to:
\[ T_{n + 1} = \binom{2n}{n} (-1)^n \]

### Step 6: Final Expression
Thus, the middle term in the expansion of \((1 - \frac{1}{x})^n (1 - x)^n\) is:
\[ \frac{(-1)^n}{x^n} \cdot \binom{2n}{n} \]

### Conclusion
The middle term in the expansion of \((1 - \frac{1}{x})^n (1 - x)^n\) is:
\[ \binom{2n}{n} \cdot (-1)^n \]