From a group of $7$ men and $6$ women,five persons are to be selected to form a committee so that at least $3$ men are there on the committee. In how many ways can it be done?

Out of $6$ books,in how many ways can a set of one or more books be chosen?

In a group of $6$ boys and $4$ girls,four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

If $^{2n}C_3 : ^nC_2 = 44:3$,then for which of the following values of $r$,the value of $^nC_r$ will be $15$?

Each of the $10$ letters $A, H, I, M, O, T, U, V, W$ and $X$ appears the same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many three-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

$A$ person wants to climb a $n$-step staircase using one step or two steps. Let $C_n$ denote the number of ways of climbing the $n$-step staircase. Then $C_{18} + C_{19}$ equals