The coefficient of $x^{13}$ in the expansion of $(1 -x)^5(1 + x + x^2 + x^3)^4$ is :-

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

The greatest coefficient in the expansion of ${(1 + x)^{2n + 2}}$ is

For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ 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

The term independent of $x$ in the expansion of ${\left( {\frac{1}{2}{x^{1/3}} + {x^{ - 1/5}}} \right)^8}$ will be