If $^{15}C_{3r} = ^{15}C_{r+3}$,then the value of $r$ is

In an examination,Yamini has to select $4$ questions from each part. There are $6, 7,$ and $8$ questions in Part-$I,$ Part-$II,$ and Part-$III,$ respectively. What is the number of possible combinations in which she can choose the questions?

In how many ways can $10$ true-false questions be replied?

If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is

$20$ persons are invited for a party. In how many different ways can they and the host be seated at a circular table,if two particular persons are to be seated on either side of the host?