There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84,$ then the value of $m$ is

Let $A_1,A_2,……..A_{11}$ are players in a team with their T-shirts numbered $1,2,…..11$. Hundred gold coins were won by the team in the final match of the series. These coins is to be distributed among the players such that each player gets atleast one coin more than the number on his T-shirt but captain and vice captain get atleast $5$ and $3$ coins respectively more than the number on their respective T-shirts, then in how many different ways these coins can be distributed ?

If $^{n} C_{8}=\,^{n} C_{2},$ find $^{n} C_{2}.$

In how many ways can one select a cricket team of eleven from $17$ players in which only $5$ players can bowl if each cricket team of $11$ must include exactly $4$ bowlers?

The number of ways in which we can select three numbers from $1$ to $30$ so as to exclude every selection of all even numbers is