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

$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is:

Let $n$ and $k$ be positive integers such that $n \ge \frac{k(k + 1)}{2}$. The number of solutions $(x_1, x_2, ..., x_k)$ where $x_1 \ge 1, x_2 \ge 2, ..., x_k \ge k$ are all integers,satisfying $x_1 + x_2 + ... + x_k = n$,is

If ${ }^{2n}C_3 : { }^{n}C_3 = 12 : 1$,then $n =$

In how many ways can two balls of the same color be selected from $4$ distinct black balls and $3$ distinct white balls?

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