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

The least value of the natural number $n$ satisfying $C(n, 5)+C(n, 6)>C(n+1, 5)$ is

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

The number of ways to select one or more books from $6$ books is:

Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples,$5$ white apples,and $8$ oranges. If in the selected $5$ fruits,at least $2$ oranges,at least $1$ red apple,and at least $1$ white apple must be included,then the number of ways Anil's mother can offer $5$ fruits to Anil is $........$

If $^{20}C_{n+2} = ^nC_{16}$,then the value of $n$ is