In how many ways can one girl and one boy be selected from a group of $15$ boys and $8$ girls?

The number of ways in which $15$ identical gold coins can be distributed among $3$ persons such that each one gets at least $3$ gold coins is

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:

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

In a lottery,a person chooses six different natural numbers at random from $1$ to $20$. If these six numbers match the six numbers already fixed by the lottery committee,he wins the prize. What is the probability of winning the prize in the game? [Hint: The order of the numbers is not important.]

The number of ways in which $17$ apples can be distributed among $4$ guests such that each guest gets at least $3$ apples is