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

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

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 . . . . . . .

The least value of a natural number $n$ such that $\binom{n-1}{5} + \binom{n-1}{6} < \binom{n}{7}$ is:

If $1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n! = 11! - 1$,then the maximum value of ${}^n C_r$ is

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are two red cards and two black cards?