${ }^{34}C_{10} + 3 \cdot { }^{34}C_{9} + 3 \cdot { }^{34}C_{8} + { }^{34}C_{7} = $

$\sum \limits_{k=0}^{6} {}^{51-k}C_{3}$ is equal to

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

If $c_0, c_1, c_2, \ldots, c_n$ denote the coefficients in the expansion of $(1+x)^n$,then the value of $c_1 + 2c_2 + 3c_3 + \ldots + nc_n$ is

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

If $26 \left( \frac{2}{3} \binom{12}{2} + \frac{2}{5} \binom{12}{4} + \frac{2}{7} \binom{12}{6} + \dots + \frac{2}{13} \binom{12}{12} \right) = 3^{13} - \alpha$,then $\alpha$ is equal to: