$A$ group consists of $8$ boys and $5$ girls. The number of committees of $5$ persons that can be formed,if the committee consists of at least $2$ girls and at most $2$ boys,is:

If ${ }^{n-3} C_r + B \cdot { }^{n-3} C_{r-1} + B^{\prime} \cdot { }^{n-3} C_{r-2} + { }^{n-3} C_{r-3} = { }^n C_r$ holds for all $n \geq r \geq 3$,then $(B, B^{\prime}) = $.

In an election,there are $8$ candidates,out of which $5$ are to be chosen. If a voter may vote for any number of candidates but not greater than the number to be chosen,then in how many ways can a voter vote?

Let $S_r = \{(x, y, z) : x + y + z = 11, x \geq r, y \geq r, z \geq r, x, y, z, r \in \mathbb{Z}\}$ and $n(S_r)$ represents the number of elements in $S_r$. Then $n(S_2) + n(S_3) + n(S_4) = $

If $\binom{2n}{3} : \binom{n}{2} = 44 : 3$ and $\binom{n}{r} = 15$,then $r = \dots$

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements is: