Let $N$ be the set of positive integers. The number of distinct triplets $(x, y, z)$ satisfying $x, y, z \in N, x < y < z$ and $x+y+z=12$ is

In how many ways can $8$ identical balls be distributed into $3$ distinct boxes such that no box remains empty?

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements is:

If $^{15}C_{3r} = ^{15}C_{r+3}$,find the value of $r$.

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

$A$ committee of $4$ members is to be formed from $4$ couples (husbands and wives). In how many ways can the committee be formed such that no couple is included in the committee?