If $^nC_{12} = ^nC_6$,then $^nC_2 = $

$A$ committee of $4$ members is to be formed from $4$ couples (husbands and wives). In how many ways can the committee be formed such that no couple is included in the committee?

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

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

If ${}^nC_{r-1}=330$,${}^nC_r=462$,and ${}^nC_{r+1}=462$,then $r$ is equal to

Determine the number of $5$-card combinations out of a deck of $52$ cards if each selection of $5$ cards has exactly one king.