$A$ group consists of $8$ boys and $5$ girls. The number of committees of $5$ persons that can be formed,if the committee consists of at least $2$ girls and at most $2$ boys,is:

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men such that it contains at least $5$ women. Find the number of ways the committee can be formed such that women are in the majority and the number of ways the committee can be formed such that men are in the majority,respectively.

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2$ and $z \geq 5$ is:

The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

In how many ways can a team of $3$ boys and $3$ girls be selected from $5$ boys and $4$ girls?

In how many ways can a committee of $6$ members be formed out of $10$ members,such that it always includes a specified member?