There are $5$ apples,$4$ mangoes,$3$ oranges,and $1$ each of $2$ other varieties of fruits. The number of ways of selecting at least one fruit of each kind is

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals

In how many ways can a committee of $6$ members be formed out of $10$ members,such that it always includes a specified member?

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

$A$ 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

The number of elements in the set $\{x \in N : ^{20-2x}C_{x-3} \in N\}$ is