The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is

$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee consists of at most $3$ girls?

The number of ways to select $n$ objects from $2n$ objects, of which $n$ are identical and the rest are different, is:

The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

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