Question

The greatest coefficient in the expansion of ${\left(x+\frac{1}{x}\right)}^{2n}$ is

• A. $\frac{1\cdot 3\cdot 5\dots (2n-1)\cdot 2^n}{n!}$
• B. $\frac{2n!}{(n!{)}^2}$
• C. $\frac{n!}{{\left[\left(\frac{n}{2}\right)!\right]}^2}$
• D. none of these

Answer 1

Answer: (A), (B)

Since the middle term has greatest coefficient,
$\therefore$ greatest coefficient $=$ coefficient of the middle term
$={}^{2n}{C}_n=\frac{(2n)!}{n!n!}$
$=\frac{2n(2n-1)(2n-2)(2n-3)\dots 4\cdot 3\cdot 2\cdot 1}{n!n!}$
$=\frac{[(2n-1)(2n-3)\dots 3\cdot 1][2n(2n-2)(2n-4)\dots 4\cdot 2]}{n!n!}$
$=\frac{[1\cdot 3\cdot 5\dots (2n-1)]2^n[n(n-1)(n-2)\dots 2\cdot 1]}{n!n!}$
$=\frac{1\cdot 3\cdot 5\dots (2n-1)2^{n}n!}{n!n!}=\frac{1\cdot 3\cdot 5\dots (2n-1)2^n}{n!}$