If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

In a touring cricket team,there are $16$ players in all,including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players can be chosen from these,such that the team includes exactly $3$ bowlers and $1$ wicket-keeper?

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

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?

There are $10$ points in a row. In how many ways can $4$ points be selected such that no two of them are consecutive?

$A$ question paper consists of two sections having $3$ and $4$ questions respectively. The following note is given on the paper: "It is not necessary to attempt all the questions. One question from each section is compulsory". In how many ways can a candidate select the questions?