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

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men. The number of committees in which the women are in majority is

If $x, y$ and $r$ are positive integers,then $^x{C_r} + ^x{C_{r-1}} ^y{C_1} + ^x{C_{r-2}} ^y{C_2} + \dots + ^y{C_r} = $

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 . . . . . . .

$A$ committee of $5$ is to be formed out of $6$ men and $4$ ladies. The number of ways this can be done,when at most $2$ ladies are included,is

In how many ways can $5$ red and $4$ white balls be drawn from a bag containing $10$ red and $8$ white balls?