The number of ways in which thirty five apples can be distributed among $3$ boys so that each can have any number of apples, is

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$, if all digits are to appear in the same number at least once, is

A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750$, then $n$ is equal to

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

$^n{C_r}{ + ^{n - 1}}{C_r} + ......{ + ^r}{C_r}$ =

If $^{15}{C_{3r}}{ = ^{15}}{C_{r + 3}}$, then the value of $r$ is