Number of rational terms in the expansion of ${\left( {\sqrt 2 \,\, + \,\,\sqrt[4]{3}} \right)^{100}}$ is :

In the expansion of $(1+a)^{m+n},$ prove that coefficients of $a^{m}$ and $a^{n}$ are equal.

In the expansion of ${\left( {x - \frac{3}{{{x^2}}}} \right)^9},$ the term independent of $x$ is

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+m x)^{50}$ is $(3 n+1)^{51} C_3$ for some positive integer $n$. Then the value of $n$ is

In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to

The coefficient of $t^{50}$ in $(1 + t^2)^{25}(1 + t^{25})(1 + t^{40})(1 + t^{45})(1 + t^{47})$ is -