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

How many words of $4$ letters beginning with '$A$' or '$E$' can be formed with the letters of the word '$EQUATOR$'?

Consider three boxes,each containing $10$ balls labelled $1, 2, dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is:

There are $3$ candidates for a post and one is to be selected by the votes of $7$ men. The number of ways in which votes can be given is

In how many ways can $5$ prizes be distributed among four students when every student can take one or more prizes?

Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar is connected by beams to the top of all its non-adjacent pillars,then the total number of beams is: