The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is $....$

If the coefficient of $4^{th}$ term in the expansion of ${(a + b)^n}$ is $56$, then $n$ is

Coefficient of $x$ in the expansion of ${\left( {{x^2} + \frac{a}{x}} \right)^5}$ is

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

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

The number of integral terms in the expansion of ${({5^{1/2}} + {7^{1/6}})^{642}}$ is