If the coefficients of $x^5$ and $x^6$ are equal in the expansion of $(a+\frac{x}{5})^{65}$,then the coefficient of $x^2$ in the expansion of $(a+\frac{x}{5})^4$ is

If $p$ and $q$ are respectively the coefficients of $x^{-3}$ and $x^{-5}$ in the expansion of $\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{21}, x > 0$,then $\frac{5p}{4q} = $

If the coefficient of $x^3$ in the binomial expansion of $x^3(2 \sqrt{3} x^2 + \frac{1}{kx})^{12}$ is $880$,then $k$ is equal to

If the coefficients of $3$ consecutive terms in the expansion of $(1+x)^{23}$ are in arithmetic progression,then those terms are

If the coefficients of the $r$-th,$(r+1)$-th,and $(r+2)$-th terms in the expansion of $(1+x)^n$ are respectively in the ratio $2:4:5$,then $(r, n) =$

Arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then,the number of terms in the expansion having integer power of $x$ is