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

If ${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$,then the minimum value of $n$ is

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) = $

In a Mathematics examination,there are $20$ questions of equal marks. The question paper is divided into three sections: $A, B$,and $C$. $A$ student is required to attempt a total of $15$ questions,taking at least $4$ questions from each section. If section $A$ has $8$ questions,section $B$ has $6$ questions,and section $C$ has $6$ questions,then the total number of ways a student can select $15$ questions is:

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these do the four cards belong to four different suits?

In how many ways can a committee of $6$ members be formed from $8$ men and $4$ women such that the committee contains at least $3$ women?