If ${S_n} = \sum\limits_{r = 0}^n {\frac{1}{{^n{C_r}}}} $ and ${t_n} = \sum\limits_{r = 0}^n {\frac{r}{{^n{C_r}}}} $,then $\frac{{{t_n}}}{{{S_n}}}$ is equal to

$\sum_{k=0}^{20} \left({}^{20}C_{k}\right)^{2}$ is equal to :

If $C_r = { }^n C_r$,then find the sum $C_0 + C_4 + C_8 + C_{12} + \ldots$

For $z \in \mathbb{C}$,if $(1+z)^n = 1 + { }^n C_1 z + { }^n C_2 z^2 + \ldots + { }^n C_n z^n$ and $\sum_{r=0}^{100} { }^{100} C_r \sin(rx) = \left(2 \cos \frac{x}{2}\right)^{100} \sin(kx)$,then $k =$

$C_0 - C_1 + C_2 - C_3 + \dots + (-1)^n C_n$ is equal to

If $(\frac{1}{^{15}C_{0}}+\frac{1}{^{15}C_{1}})(\frac{1}{^{15}C_{1}}+\frac{1}{^{15}C_{2}})...(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}) = \frac{a^{13}}{^{14}C_{0} \cdot ^{14}C_{1} \cdot ... \cdot ^{14}C_{12}}$, then $30a$ is equal to: