$A$ 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?

The number of ordered triplets of positive integers which are solutions of the equation $x + y + z = 100$ is

$_n{P_r} \div \binom{n}{r} = ..........$

There are $6$ different novels and $3$ different poetry books on a table. If $4$ novels and $1$ poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle,then the number of such possible arrangements is

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

Everybody in a room shakes hands with everybody else. The total number of handshakes is $45$. The total number of persons in the room is