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

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

If ${ }^{n+4} C_{n+1}-{ }^{n+3} C_n=15(n+2)$,then $n=$

The greatest integer which divides $(p+1)(p+2)(p+3) \ldots (p+q)$ for all $p \in N$ and fixed $q \in N$ is

In how many ways can a team of $11$ players be formed out of $25$ players,if $6$ out of them are always to be included and $5$ are always to be excluded?

$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee must consist of exactly $3$ girls?