Question

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

• A. ${}^{2n}{C}_n$
• B. ${}^{-2n}{C}_n$
• C. ${}^{-2n}{C}_{n-1}$
• D. none of these

Answer 1

The correct option is A ${}^{2n}{C}_n$

$(1-\frac{1}{x}{)}^n(1-x{)}^n$
$(\frac{x-1}{x}{)}^n(1-x{)}^n$
Case I - $n$ is even.
Hence the above expression reduces to
$\frac{1}{x^n}(1-x{)}^{2n}$
The middle term will be $(n+1{)}^{th}$ term.
Hence the coefficient will be.
$T_{n+1}={}^{2n}{C}_n$
Case II - $n$ is odd.
Then the above expression reduces to.
$-\frac{1}{x^n}(1-x{)}^{2n}$
Middle term will be $(n+1{)}^{th}$ term.
Hence coefficient will be
$T_{n+1}=-\left(\right(-1{)}^n{}^{2n}{C}_n)={}^{2n}{C}_n$