Find the mean of the values $0, 1, 2, \dots, n$ with respective weights $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})^{n}$,in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}: 1$. If the sixth term from the beginning is $\frac{\alpha}{\sqrt[4]{3}}$,then $\alpha$ is equal to $.......$

The two middle terms in the expansion of ${\left( {x - \frac{1}{x}} \right)^{11}}$ are:

The positive value of $a$ such that the coefficient of $x^5$ is equal to that of $x^{15}$ in the expansion of $(x^2 + \frac{a}{x^3})^{10}$ is

The independent term in the expansion of $(1+x+2x^2)(\frac{3x^2}{2}-\frac{1}{3x})^9$ is

The coefficients of three successive terms in the expansion of $(1 + x)^n$ are $165, 330$ and $462$ respectively. Then the value of $n$ is: