The positive value of $a$ so that the co-efficient of $x^5$ is equal to that of $x^{15}$ in the expansion of ${\left( {{x^2}\,\, + \,\,\frac{a}{{{x^3}}}} \right)^{10}}$ is
$\frac{1}{{2\,\sqrt 3 }}$
$\frac{1}{{\sqrt 3 }}$
$1$
$2 \sqrt 3$
Find the cocfficient of $a^{5} b^{7}$ in $(a-2 b)^{12}$
The coefficient of ${x^{32}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ is
The coefficient of ${x^{39}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ is
The greatest coefficient in the expansion of ${(1 + x)^{2n + 1}}$ is
The coefficient of ${x^{100}}$ in the expansion of $\sum\limits_{j = 0}^{200} {{{(1 + x)}^j}} $ is