If the coefficients of the $(2r + 3)^{th}$ and $(r - 1)^{th}$ terms in the expansion of $(1 + x)^{15}$ are equal,then the value of $r$ is:

The numerically greatest term in the expansion of $(3x - 16y)^{15}$ when $x = \frac{2}{3}$ and $y = \frac{3}{2}$ is

Find the $r^{\text{th}}$ term from the end in the expansion of $(x+a)^{n}$.

Find the coefficient of $x^{5}$ in the expansion of $(x+3)^{8}$.

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:

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $(1+\alpha x+\beta x^2)(1+x)^{11}$ are $396$ and $144$ respectively,then $\alpha^2+\beta^2=$