If a committee consists of $3$ men and $2$ women,in how many ways can a committee of $5$ members be formed from $5$ men and $4$ women?

If $^{15}C_{3r} = ^{15}C_{r+3}$,find the value of $r$.

In how many ways can $5$ letters be selected from the letters of the word $INDEPENDENT$?

$A$ 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?

If the domain and range of $f(x) = ^{9-x}C_{x-1}$ contain $m$ and $n$ elements respectively,then:

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals