If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $

If ${ }^{n} C_{12}={ }^{n} C_{8}$,then $n$ is equal to

The number of integers between $100$ and $1000$ such that the sum of their digits is equal to $14$ is:

The number of ways to distribute $25$ identical apples among $4$ boys,such that each boy receives at least $4$ apples,is

To fill $12$ vacancies,there are $25$ candidates,of which $5$ are from the scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all,then the number of ways in which the selection can be made is:

If ${}^n C_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression ${}^n C_{r+1} + {}^n C_{r-1} + 2{}^n C_r$ equals