If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is
$\frac{{(2n)!}}{{(m)!\,(2n - m)!}}$
$\frac{{(2n)!\,3!\,3!}}{{(2n - m)!}}$
$\frac{{(2n)!}}{{\left( {\frac{{2n - m}}{3}} \right)\,!\,\left( {\frac{{4n + m}}{3}} \right)\,!}}$
None of these
If for positive integers $r > 1,n > 2$ the coefficient of the ${(3r)^{th}}$ and ${(r + 2)^{th}}$ powers of $x$ in the expansion of ${(1 + x)^{2n}}$ are equal, then
If $1 + {x^4} + {x^5} = \sum\limits_{i = 0}^5 {{a_i}\,(1 + {x})^i,} $ for all $x$ in $R,$ then $a_2$ is
The coefficient of ${x^{100}}$ in the expansion of $\sum\limits_{j = 0}^{200} {{{(1 + x)}^j}} $ is
Find $a$ if the $17^{\text {th }}$ and $18^{\text {th }}$ terms of the expansion ${(2 + a)^{{\rm{50 }}}}$ are equal.
The coefficient of ${x^n}$in expansion of $(1 + x)\,{(1 - x)^n}$ is