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

If ${ }^9 C_3+{ }^9 C_5={ }^{10} C_r$ for some $r \in N$,then $r=$

There are ten boys $B_{1}, B_{2}, \ldots, B_{10}$ and five girls $G_{1}, G_{2}, \ldots, G_{5}$ in a class. The number of ways of forming a group consisting of three boys and three girls,such that $B_{1}$ and $B_{2}$ are not both members of the same group,is

Sangeeta plans a dinner party for $6$ guests. Out of $10$ friends,in how many ways can she choose the guests if two specific friends cannot attend the party together?

In how many ways can one girl and one boy be selected from a group of $15$ boys and $8$ girls?

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