The number of integral solutions to the equation $x+y+z=21$,where $x \geq 1, y \geq 3, z \geq 4$,is equal to $..........$.

To fill $12$ vacancies,there are $25$ candidates,of which $5$ are from the scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all,then the number of ways in which the selection can be made is:

The number of ways in which a cricket team of $11$ members can be formed out of $6$ batsmen,$6$ bowlers,$4$ all-rounders and $4$ wicket keepers by selecting at least $4$ batsmen,at least $3$ bowlers,at least $2$ all-rounders and only $1$ wicket keeper is:

There are $25$ candidates for $12$ positions,out of which $5$ are from the reserved category. If $3$ positions are reserved and the remaining are open,in how many ways can the selection be made?

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 $12$ is to be formed from $9$ women and $8$ men such that at least $5$ women are included. Find the number of committees in which women are in the majority and the number of committees in which men are in the majority,respectively.