Question

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

• A. $0$,if $n$ is odd
• B. $(-1{)}^{n-1/2}{\cdot }^n{\text{C}}_{n-1/2}$,if $n$ is odd
• C. $(-1{)}^{n/2}{\cdot }^n{\text{C}}_{n/2}$,if $n$ is even
• D. none of these

Answer 1

Answer: (D)

none of these

$(1+x{)}^n(1+\frac{1}{x}{)}^n$
$=\frac{1}{x^n}(1+x{)}^n(1+x{)}^n$
$=\frac{1}{x^n}(1+x{)}^{2n}$
Writing the general term, we get
$T_{r+1}={}^{2n}{C}_r{x}^{r-n}$
For the term independent of x.
$r-n=0$
$r=n$
Hence the coefficient of the term independent of x will be
${}^{2n}{C}_n$