$A$ class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$,then $b + 3g$ is equal to.

If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2$ and $z \geq 5$ is:

If $\binom{n}{r-1} = 36$,$\binom{n}{r} = 84$,and $\binom{n}{r+1} = 126$,then $r = \dots$

$A$ student is allowed to select at most $n$ books from a collection of $(2n+1)$ books. If the total number of ways in which one can select at least one book is $255$,then the value of $n$ is

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