If the middle term in the expansion of ${\left( {{x^2} + \frac{1}{x}} \right)^n}$ is $924{x^6}$, then $n = $

The term independent of $x$ in ${\left( {2x - \frac{1}{{2{x^2}}}} \right)^{12}}$is

In the expansion of ${(1 + 3x + 2{x^2})^6}$ the coefficient of ${x^{11}}$ is

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n}$, 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$.......$

If the sum of the coefficients in the expansion of $(x - 2y + 3 z)^n,$ $n \in N$ is $128$ then the greatest coefficie nt in the exp ansion of $(1 + x)^n$ is

Number of rational terms in the expansion of ${\left( {{3^{\frac{1}{8}}} + {5^{\frac{1}{3}}}} \right)^{400}}$ is