Question

Find the term independent of \( x \) in the expansion of \( (1 + x)^{n} (1 + \frac{1}{x})^{n} \).

• A. \(\binom{2n}{n}\)
• B. \( ^{n}C_{n} \)
• C. \( (^{n}C_{n/2})^{2} \)
• D. \( (^{n}C_{n})^{2} \)

Hint:

Whenever you see \((1+x)^n(1+1/x)^n\), it is mathematically identical to \(\frac{(1+x)^{2n}}{x^n}\). This trick quickly converts a product into a single binomial expansion problem.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A term independent of \( x \) is a term where the total power of \( x \) is zero. We first simplify the given expression into a single binomial form.

Step 2: Key Formula or Approach
1. Simplify \( (1 + 1/x)^n \). 2. Use the general term formula for \( (1+x)^N \), which is \( T_{r+1} = \binom{N}{r} x^r \).

Step 3: Detailed Explanation
1. Rewrite the second bracket: \[ (1 + 1/x)^n = \left( \frac{x + 1}{x} \right)^n = \frac{(1 + x)^n}{x^n} \] 2. Combine the terms: \[ (1 + x)^n \cdot \frac{(1 + x)^n}{x^n} = \frac{(1 + x)^{2n}}{x^n} \] 3. To find the term independent of \( x \), we need the coefficient of \( x^n \) in the numerator \( (1 + x)^{2n} \), because \( \frac{x^n}{x^n} = x^0 \). 4. The coefficient of \( x^r \) in \( (1+x)^N \) is \( \binom{N}{r} \). 5. Here, \( N = 2n \) and \( r = n \). Thus, the coefficient is \( \binom{2n}{n} \).

Step 4: Final Answer
The term independent of \( x \) is \( \binom{2n}{n} \).