$^nC_{r+1} + ^nC_{r-1} + 2^nC_r = \dots$

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

If $^{2n}C_2 : ^nC_2 = 9:2$ and $^nC_r = 10$,then $r = $

The number of $5$ card combinations out of a deck of $52$ cards such that there is exactly one ace in each combination is:

$A$ student is allowed to select at most $n$ books from a collection of $(2n+1)$ books. If the total number of ways in which one can select at least one book is $255$,then the value of $n$ is

If ${}^{10}P_r = 604800$ and ${}^{10}C_r = 120$,then $r =$