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

$^{14}{C_4} + \sum\limits_{j = 1}^4 {^{18 - j}{C_3}} $ is equal to

Number of positive integral solution of the equation $xyz = 90$ is equal to :-

In an examination, a question paper consists of $12$ questions divided into two parts i.e., Part $\mathrm{I}$ and Part $\mathrm{II}$, containing $5$ and $7$ questions, respectively. A student is required to attempt $8$ questions in all, selecting at least $3$ from each part. In how many ways can a student select the questions?

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

If $^{{n^2} - n}{C_2}{ = ^{{n^2} - n}}{C_{10}}$, then $n = $