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

In how many ways can $10$ balls be divided between two boys,such that one receives $2$ balls and the other receives $8$ balls?

$A$ committee of $12$ is to be formed from $9$ women and $8$ men such that at least $5$ women are included. Find the number of committees in which women are in the majority and the number of committees in which men are in the majority,respectively.

The number of elements in the set $S = \{(r, k) : k \in Z \text{ and } ^{36}C_{r+1} = \frac{6(^{35}C_r)}{(k^2-3)}\}$ is:

The solution set of $^{10}C_{x-1} > 2 \cdot ^{10}C_x$ is

If $\binom{n-1}{4}$,$\binom{n-1}{5}$,and $\binom{n-1}{6}$ are in arithmetic progression,find $n$.