Question

If $x^{m}$ occurs in the expansion of $(x + 1/x^{2})^{2n}$, then the coefficient of $x^{m}$ is

• A. $\frac{\left(2n\right)!}{\left(m\right)!\left(2n-m\right)!}$
• B. $\frac{\left(2n\right)!3!3!}{\left(2n-m\right)!}$
• C. $\frac{\left(2n\right)!}{\left(\frac{2n-m}{3}\right)!\left(\frac{4n+m}{3}\right)!}$
• D. none of these

Answer 1

Answer: (C)

$T_{r+1} = {^{2n}}C_{r}x^{2n-r}(\frac{1}{x^{2}})^{r}= {^{2n}}C_{r}x^{2n-3r}$
This contains $x^{m}$. If $2n -3r =m,$ then
$r=\frac{2n-m}{3}$
$\Rightarrow $ Coefficient of $x^{m} = {^{2n}}C_{r}$
$r=\frac{2n-m}{3}$
$=\frac{2n!}{(2n-r)!r!}=\frac{2n!}{(2n-\frac{2n-m}{3})!(\frac{2n-m}{3})!}$
$=\frac{2n!}{(\frac{4n+m}{3})!(\frac{2n-m}{3})!}$