If $k$ is the coefficient of $x^5$ in the expansion of $(2x^2 - \frac{1}{3x^3})^5$,then $\frac{3k}{2} = $

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$, where $\alpha \neq 0$, are equal, then the value of $\alpha$ is:

Let the $9^{\text{th}}$ term in the binomial expansion of $(3+6x)^{n}$,in the increasing powers of $6x$,be the greatest for $x=\frac{3}{2}$. If $n_{0}$ is the least value of $n$ for which this holds,and $k$ is the ratio of the coefficient of $x^{6}$ to the coefficient of $x^{3}$,then find the value of $k + n_{0}$.

If $(1+x)^{15}=a_0+a_1 x+\ldots+a_{15} x^{15}$,then $\sum_{r=1}^{15} r \frac{a_r}{a_{r-1}}$ is equal to

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $(1+\alpha x+\beta x^2)(1+x)^{11}$ are $396$ and $144$ respectively,then $\alpha^2+\beta^2=$

The coefficient of $t^{50}$ in $(1 + t^2)^{25}(1 + t^{25})(1 + t^{40})(1 + t^{45})(1 + t^{47})$ is