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

If the ratio $C(2n, 3) : C(n, 3)$ is equal to $11 : 1$,find $n$.

$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $

Let $A = \{a_1, a_2, a_3, ..., a_n\}$ be a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are formed independently. The number of ways in which these subsets can be formed such that $(P - Q)$ contains exactly $2$ elements is:

How many numbers of four digits can be formed with the digits $1, 2, 3, 4$ and $5$?

In how many ways can $7$ men and $7$ women be seated around a round table such that no two women sit together?