In how many ways can a committee of $5$ members be formed from $6$ men and $4$ women such that the committee contains at least one woman?

$A$ group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least one boy and one girl?

If $\binom{n}{3} + \binom{n}{4} > \binom{n+1}{3}$,then:

If $^{2n}C_3 : ^nC_2 = 44:3$,then for which of the following values of $r$,the value of $^nC_r$ will be $15$?

$A$ team of $5$ students is to be selected from $12$ students. If two particular students are to be included in that team,then the number of ways that such team can be selected 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