The number of all possible ways of distributing $8$ identical apples among $3$ persons is

Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is . . . . . . .

$A$ bag contains $5$ red marbles,$4$ black marbles,and $3$ white marbles. The number of ways in which $4$ marbles can be drawn so that at most $2$ of them are red is:

If a committee consists of $3$ men and $2$ women,in how many ways can a committee of $5$ members be formed from $5$ men and $4$ women?

If $20$ identical books are to be distributed among $4$ persons such that each person receives at least one book,in how many ways can this be done?

If ${}^nC_{r-1}=36$,${}^nC_r=84$,and ${}^nC_{r+1}=126$,then the value of $nr^2$ is