If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

If ${ }^{15} C_4+{ }^{15} C_5+{ }^{16} C_6+{ }^{17} C_7+{ }^{18} C_8={ }^{19} C_{r}$,then the value of $r$ is equal to

If $n$ objects are arranged in a row,then the number of ways of selecting three of these objects such that no two of them are next to each other is:

The number of elements in the set $S = \{(r, k) : k \in Z \text{ and } ^{36}C_{r+1} = \frac{6(^{35}C_r)}{(k^2-3)}\}$ is:

If ${}^n C_7 = {}^n C_6$,then ${}^n C_2 = $