An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then the number of ways in which $4$ marbles can be drawn so that at the most three of them are red is

A set contains $(2n + 1)$ elements. The number of sub-sets of the set which contains at  most $n$ elements is :-

The number of ways in which a committee of $6$ members can be formed from $8 $ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

If $^{n + 1}{C_3} = 2{\,^n}{C_2},$ then $n =$

Team $'A'$ consists of $7$ boys and $n$ girls and Team $'B'$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to

The number of ways of choosing $10$ objects out of $31$ objects of which $10$ are identical and the remaining $21$ are distinct, is