In the expansion of the following expression $1 + (1 + x) + {(1 + x)^2} + ..... + {(1 + x)^n}$ the coefficient of ${x^k}(0 \le k \le n)$ 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 coefficient of ${x^n}$in expansion of $(1 + x)\,{(1 - x)^n}$ is

Prove that the coefficient of $x^{n}$ in the expansion of $(1+x)^{2n}$ is twice the coefficient of $x^{n}$ in the expansion of $(1+x)^{2 n-1}$

If the coefficients of $x^7$ & $x^8$ in the expansion of ${\left[ {2\,\, + \,\,\frac{x}{3}} \right]^n}$ are equal , then the value of $n$ is :

The coefficient of $x^{-5}$ in the binomial expansion of ${\left( {\frac{{x + 1}}{{{x^{\frac{2}{3}}} - {x^{\frac{1}{3}}} + 1}} - \frac{{x - 1}}{{x - {x^{\frac{1}{2}}}}}} \right)^{10}}$ where $x \ne 0, 1$ , is