The coefficient of `x^(-n) " in " (1+ x)^(n) (1+ (1)/(x))^(n)` is

Options

A. 0
B. 1
C. `2^(n)`
D. 2n

Answer

To find the coefficient of \( x^{-n} \) in the expression \( (1 + x)^n \left( 1 + \frac{1}{x} \right)^n \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1 + x)^n \left( 1 + \frac{1}{x} \right)^n \]
We can rewrite \( \left( 1 + \frac{1}{x} \right)^n \) as:
\[ \left( \frac{x + 1}{x} \right)^n = \frac{(1 + x)^n}{x^n} \]
Thus, the entire expression becomes:
\[ (1 + x)^n \cdot \frac{(1 + x)^n}{x^n} = \frac{(1 + x)^{2n}}{x^n} \]

### Step 2: Expand the expression
Now we need to expand \( \frac{(1 + x)^{2n}}{x^n} \):
\[ \frac{(1 + x)^{2n}}{x^n} = (1 + x)^{2n} \cdot x^{-n} \]
This means we need to find the coefficient of \( x^{-n} \) in the expansion of \( (1 + x)^{2n} \).

### Step 3: Identify the term contributing to \( x^{-n} \)
In the expansion of \( (1 + x)^{2n} \), the general term is given by:
\[ \binom{2n}{k} x^k \]
We want the term where \( k - n = -n \) or \( k = 0 \). Thus, we need the coefficient of \( x^0 \) in \( (1 + x)^{2n} \).

### Step 4: Calculate the coefficient
The coefficient of \( x^0 \) in \( (1 + x)^{2n} \) is:
\[ \binom{2n}{0} = 1 \]

### Step 5: Conclusion
Therefore, the coefficient of \( x^{-n} \) in the original expression is:
\[ \boxed{1} \]