Compute $\frac{8!}{6! \times 2!}$

If $^{n + 1}C_3 = 2 \cdot ^nC_2$,then $n =$

If $^nC_{15} = ^nC_8$,then what is the value of $^nC_{21}$?

There are $15$ people at a party and each person shakes hands with every other person. What is the total number of handshakes?

$^{47}C_4 + \sum_{r=1}^5 {}^{52-r}C_3 = $

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: