If ${}^n C_7 = {}^n C_6$,then ${}^n C_2 = $

In an examination,a question paper consists of $12$ questions divided into two parts,i.e.,Part $I$ and Part $II$,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?

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

The number of ways in which an examiner can assign $30$ marks to $8$ questions,giving not less than $2$ marks to any question,is

Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is . . . . . . .

In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?