Question

If x^m occurs in the expansion of $\left( x + \frac{1}{x^{2}} \right)^{2n}$, then the co-efficient of x^m is

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

Answer 1

Answer: (A)

$\frac{(2n)!}{\left( \frac{2n - m}{3} \right)!\left( \frac{4n + m}{3} \right)!}$

Answer 2

General term in the expansion is $2nC_{r}x^{2n - r}\frac{1}{x^{2r}}$

= $2nC_{r}x^{2n - 3r}$

For x^m, 2n − 3r = m ⇒ $r = \frac{2n - m}{3}$

so that the coefficient of x^m is $2nC_{\frac{2n - m}{3}}$