The number of ways to give away $25$ apples to $4$ boys, each boy receiving at least $4$ apples, are

Determine the number of $5 -$ card combinations out of a deck of $52$ cards if each selection of $5$ cards has exactly one king.

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

$\sum \limits_{ k =0}^6{ }^{51- k } C _3$ is equal to