If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

If $^nC_{15} = ^nC_8$,then what is the value of $^nC_{21}$?

$A$ question paper is divided into two parts,$A$ and $B$,each consisting of $5$ questions. If a candidate has to select $6$ questions in total,in how many ways can they do so if they must select at least $2$ questions from each part?

In an election,there are $5$ candidates and $3$ vacancies. $A$ voter can vote for a maximum of $3$ candidates. In how many ways can a voter cast their vote?

The number of ways of forming the ordered pairs $(p, q)$ such that $p > q$ by choosing $p$ and $q$ from the first $50$ natural numbers 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: