If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3}\right)^{n}$ are equal, then the value of $n$ is equal to $.....$
$44$
$55$
$48$
$61$
If the coefficient of $x$ in the expansion of ${\left( {{x^2} + \frac{k}{x}} \right)^5}$ is $270$, then $k =$
If $\alpha$ and $\beta$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of
$(\mathrm{x}+\sqrt{\mathrm{x}^{2}-1})^{6}+(\mathrm{x}-\sqrt{\mathrm{x}^{2}-1})^{6}$, then
The coefficient of $\frac{1}{x}$ in the expansion of ${\left( {1 + x} \right)^n}{\left( {1 + \frac{1}{x}} \right)^n}$ is :-
The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${\left( {{x^2} + \frac{2}{x}} \right)^{15}}$ is
The term independent of $x$ in the expansion of ${\left( {\sqrt {\frac{x}{3}} + \frac{3}{{2{x^2}}}} \right)^{10}}$ will be