The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $(x^2 + \frac{2}{x})^{15}$ is

If the coefficients of the three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1:5:20$,then the coefficient of the fourth term is $............$.

For a positive integer $n$,$(1+\frac{1}{x})^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio $2:5:12$,then $n$ is equal to

If the number of integral terms in the expansion of $(3^{\frac{1}{2}} + 5^{\frac{1}{8}})^n$ is exactly $33$,then the least value of $n$ is

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n, n \in N$,if the ratio of the $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\frac{1}{6}$,then the value of ${}^n C_3$ is:

If the sum of the coefficients of all the positive powers of $x$ in the binomial expansion of $(x^{n} + \frac{2}{x^{5}})^{7}$ is $939$,then the sum of all the possible integral values of $n$ is