Everybody in a room shakes hands with everybody else. The total number of handshakes is $66$. The total number of persons in the room is

The number of elements in the set $\{x \in N : ^{20-2x}C_{x-3} \in N\}$ is

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

The solution set of $^{10}C_{x-1} > 2 \cdot ^{10}C_x$ 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

In how many ways can a group of $7$ persons be formed from $6$ boys and $4$ girls such that the boys are in the majority?