If the coefficient of $x^7$ in $(ax - \frac{1}{bx^2})^{13}$ and the coefficient of $x^{-5}$ in $(ax + \frac{1}{bx^2})^{13}$ are equal,then $a^4 b^4$ is equal to :

The coefficient of $x^{n-6}$ in the expansion $n! \left[ x - \left( \frac{^nC_0 + ^nC_1}{^nC_0} \right) \right] \left[ \frac{x}{2} - \left( \frac{^nC_1 + ^nC_2}{^nC_1} \right) \right] \left[ \frac{x}{3} - \left( \frac{^nC_2 + ^nC_3}{^nC_2} \right) \right] \dots \left[ \frac{x}{n} - \left( \frac{^nC_{n-1} + ^nC_n}{^nC_{n-1}} \right) \right]$ is equal to:

If the coefficient of $x$ in the expansion of $\left(x^2+\frac{k}{x}\right)^5$ is $270$,then $k$ is equal to

If the ratio of the $7^{th}$ term from the beginning to the $7^{th}$ term from the end in the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n$ is $\frac{1}{6}$,then $n=$

If the coefficients of the second,third,and fourth terms in the expansion of $(1 + x)^{2n}$ are in $A.P.$,then $2n^2 - 9n + 7$ is equal to

In the expansion of $(1+x)^n$,the coefficients of the $p^{th}$ and $(p+1)^{th}$ terms are respectively $p$ and $q$. Then $p+q$ is equal to: