Question

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

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

Answer 1

Answer: (A), (B)

$\frac{\left(2n\right)!}{(n!{)}^2}$
$\frac{1\cdot 3\cdot \mathrm{5...}(2n-1)}{n!}{2}^n$

Since the middle term has greatest coefficients,

$\therefore$ greatest coefficient $=$ coefficients of the middle term

${=}^{2n}{C}_n=\frac{\left(2n\right)!}{n!n!}$

$=\frac{2n(2n-1)(2n-2)(2n-3)...4.3.2.1}{n!n!}$

$=\frac{[(2n-1)(2n-3)...3.1][2n(2n-2)(2n-4)...4.2]}{n!n!}$

$=\frac{\left[\mathrm{1.3.5...}(2n-1)\right]2^n[n(n-1)(n-2)...2.1]}{n!n!}$

$=\frac{\mathrm{1.3.5...}(2n-1)2^{n}n!}{n!n!}=\frac{\mathrm{1.3.5...}(2n-1)2^n}{n!}$