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

If the total number of $m$-element subsets of the set $A = \{a_{1}, a_{2}, \ldots, a_{n}\}$ is $k$ times the number of $m$-element subsets containing $a_{4}$, then $n$ is

$A$ man has $7$ friends. In how many ways can he invite one or more of them for a tea party?

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

The number of ways to distribute $25$ identical apples among $4$ boys,such that each boy receives at least $4$ apples,is

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 $..........$.