$A$ bag contains $5$ black and $6$ red balls. Determine the number of ways in which $2$ black and $3$ red balls can be selected.

$A$ student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select at least one book is $63$,then the value of $n$ is

The number of ways of selecting a committee of $30$ persons from $20$ boys,$20$ girls,and $20$ teachers such that the participation of the number of boys,girls,and teachers in that committee is equal,is

The number of values of $r$ satisfying $^{69}C_{3r-1} - ^{69}C_{r^2} = ^{69}C_{r^2-1} - ^{69}C_{3r}$ is:

Compute $\frac{12!}{10! \times 2!}$

$A$ set contains $11$ elements. The number of subsets of the set which contain at most $5$ elements is