The coefficient of ${x^n}$in expansion of $(1 + x)\,{(1 - x)^n}$ is

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

Coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

Find $a$ if the $17^{\text {th }}$ and $18^{\text {th }}$ terms of the expansion ${(2 + a)^{{\rm{50 }}}}$ are equal.

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is $....$

Let $\alpha>0, \beta>0$ be such that $\alpha^{3}+\beta^{2}=4 .$ If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)^{10}$ is $10 k$ then $\mathrm{k}$ is equal to