If $^nC_r = 84$,$^nC_{r-1} = 36$,and $^nC_{r+1} = 126$,then $n$ equals

$A$ student has to answer $10$ questions,choosing at least $4$ from each of the parts $A$ and $B$. If there are $6$ questions in part $A$ and $7$ in part $B$,then the number of ways the student can choose $10$ questions is:

If $\binom{n-1}{r} = (k^2 - 3) \binom{n}{r+1}$,then $k \in \dots$

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 $........$

The number of integers $x, y, z, w$ satisfying $x+y+z+w=25$ and $x, y, z \geq -1, w \geq 1$ is

The number of ways in which a team of $11$ players can be formed out of $25$ players,if $6$ out of them are always to be included and $5$ of them are always to be excluded,is