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

If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men such that it contains at least $5$ women. Find the number of ways the committee can be formed such that women are in the majority and the number of ways the committee can be formed such that men are in the majority,respectively.

The number of $5$ card combinations out of a deck of $52$ cards such that there is exactly one ace in each combination is:

From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is

If $^nC_4, ^nC_5,$ and $^nC_6$ are in $A.P.,$ then $n$ can be