If $\sum_{r=1}^9 \left(\frac{r+3}{2^r}\right) \cdot {}^9C_r = \alpha \left(\frac{3}{2}\right)^9 - \beta$,where $\alpha, \beta \in N$,then $(\alpha + \beta)^2$ is equal to

The numerically greatest term in the expansion of $(5+3x)^6$ when $x=1$ is:

If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is $405$,then $|k|$ equals

If $\binom{p}{q} = {}^{p}C_{q}$ and $\sum_{i=0}^{m} \binom{10}{i} \binom{20}{m-i}$ is maximum,then $m=$

Show that the middle term in the expansion of $(1+x)^{2n}$ is $\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{n!} 2^n x^n$,where $n$ is a positive integer.

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n, n \in N$,if the ratio of the $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\frac{1}{6}$,then the value of ${}^n C_3$ is: