If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

If $\binom{n-1}{r} = (k^2 - 3) \binom{n}{r+1}$,then $k \in \dots$

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,is

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

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