There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?

If $a_n = \sum_{r=0}^n \frac{1}{{}^n C_r}$,then $\sum_{r=0}^n \frac{r}{{}^n C_r} = $

$A$ man has $7$ relatives: $4$ ladies and $3$ gentlemen. His wife also has $7$ relatives: $3$ ladies and $4$ gentlemen. In how many ways can they invite a dinner party of $3$ ladies and $3$ gentlemen,such that there are $3$ of the man's relatives and $3$ of the wife's relatives?

There are $5$ volumes of Mathematics among $25$ books. They are arranged on a shelf in random order. The probability that the volumes of Mathematics stand in increasing order from left to right (the volumes are not necessarily kept side by side) is

Find the remainder when $(1!)^2 + (2!)^2 + (3!)^2 + \dots + (100!)^2$ is divided by $10^2$.

The number $(49^{2}-4)(49^{3}-49)$ is divisible by (in $!$)