If the expansion of $(y^2 + \frac{c}{y})^5$,the coefficient of $y$ is:

If the coefficients of the $(2r+6)^{\text{th}}$ and $(r-1)^{\text{th}}$ terms in the expansion of $(1+x)^{21}$ are equal,then the value of $r$ is:

If the sum of the coefficients in the expansion of $(x + y)^n$ is $1024$,then the value of the greatest coefficient in the expansion is

The sum of the rational terms in the expansion of $(\sqrt{2}+\sqrt[5]{3})^{10}$ is

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $(2+\frac{1}{\sqrt{2}})^{200}$. If $\frac{{}^{200}C_{99} K}{a} = \frac{2^{\ell} m}{n}$,where $m$ and $n$ are odd numbers,then the ordered pair $(\ell, n)$ is equal to:

The coefficient of $x^4$ in the expansion of $\frac{(1-3 x)^2}{(1-2 x)}$ is equal to