$^nC_{r+1} + ^nC_{r-1} + 2^nC_r = \dots$

Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.

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

In how many ways can $8$ identical balls be distributed into $3$ distinct boxes such that no box remains empty?

There are $12$ volleyball players in total in a college,out of which a team of $9$ players is to be formed. If the captain always remains the same,in how many ways can the team be formed?

$A$ father with $8$ children takes $3$ children to the zoo at a time. If he can go more than once without taking the same $3$ children together,in how many ways can the father go to the zoo?