The number of ways in which $3$ identical balls can be distributed into $7$ distinct bins is

If $^{20}C_{n+2} = ^nC_{16}$,then the value of $n$ is

If $\binom{n-1}{4}, \binom{n-1}{5}$ and $\binom{n-1}{6}$ are in Arithmetic Progression,then find the relation.

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

In how many ways can $10$ people be seated in $2$ boats such that each boat has $5$ people and two specific people do not sit in the same boat?

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.