Everybody in a room shakes hands with everybody else. The total number of handshakes is $45$. The total number of persons in the room is

$^nC_r + ^{n-1}C_r + ...... + ^rC_r =$

If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

There are $10$ persons named $A, B, \dots, J$. We have space to keep only $5$ of them. If $A$ must be included,and $G$ and $H$ must not be included in the group of $5$,in how many ways can we arrange the group in a row?

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:

The number of ways a committee of $8$ members can be formed from a group of $10$ men and $8$ women such that the committee contains at most $5$ men and at least $5$ women is: