In how many ways can $12$ balls be distributed between two friends such that one gets $8$ balls and the other gets $4$ balls?

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

The number of $5$ card combinations out of a deck of $52$ cards such that there is exactly one ace in each combination is:

$A$ question paper consists of two sections having $3$ and $4$ questions respectively. The following note is given on the paper: "It is not necessary to attempt all the questions. One question from each section is compulsory". In how many ways can a candidate select the questions?

The number of four-digit numbers that can be formed using all the digits from $1$ to $9$ (excluding zero) such that every number has exactly $2$ distinct digits is:

The number of ordered pairs $(r, k)$ for which $6 \cdot ^{35}C_{r} = (k^{2} - 3) \cdot ^{36}C_{r+1}$,where $k$ is an integer,is