The coefficient of $x^5$ in the expansion of $(1+x)^{21}+(1+x)^{22}+\ldots+(1+x)^{30}$ is

The term independent of $x$ in the expansion of $\left(\sqrt{x}-\frac{2}{\sqrt{x}}\right)^{18}$ 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:

The number of integral terms in the expansion of $(7^{1/3} + 11^{1/9})^{6561}$ is:

The middle term in the expansion of ${\left( {1 - \frac{1}{x}} \right)^n}{\left( {1 - x} \right)^n}$ in powers of $x$ is

The sum of all possible values of $n \in N$ such that the coefficients of $x$, $x^2$, and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$ are in arithmetic progression is: