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

If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

$A$ linguistic club of a certain Institute consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this group,including the selection of a Captain (from among these $4$ members) for the team. If the team has to include at most one boy,the number of ways of selecting the team is:

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?

$A$ group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has no girl?