If $x, y$ and $r$ are positive integers,then $^x{C_r} + ^x{C_{r-1}} ^y{C_1} + ^x{C_{r-2}} ^y{C_2} + \dots + ^y{C_r} = $

The greatest integer which divides $(p+1)(p+2)(p+3) \ldots (p+q)$ for all $p \in N$ and fixed $q \in N$ is

If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements is:

Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is . . . . . . .

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