Question

The term independent of $x$ in the expansion of ${(1+x)}^n{(1+\frac{1}{x})}^n,is$

• A. $C{{}_0}^2+2{C_1}^2+3.C{{}_2}^2+...(n+1)C{{}_n}^2$
• B. ${(C_0+C_1+...+C_n)}^2$
• C. $C{{}_0}^2+C{{}_1}^2+...+C{{}_n}^2$
• D. none of these

Answer 1

Answer: (C)

$C{{}_0}^2+C{{}_1}^2+...+C{{}_n}^2$

$(1+x{)}^n(1+\frac{1}{x}{)}^n$
$=\left(\frac{1}{x}{)}^n\right(1+x{)}^{2n}$
The coefficient of $x^n$ in $(1+x{)}^{2n}$ implies
$T_{n+1}={}^{2n}{C}_n{x}^n$
Hence the term independent of x will be $T_{n+1}$ and its coefficient will be ${}^{2n}{C}_n$