Out of $6$ boys and $4$ girls,a group of $7$ is to be formed. In how many ways can this be done if the group is to have a majority of boys?

$A$ 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

If the best and the worst paper never appear together,then six examination papers can be arranged in how many ways?

In how many ways can four children be made to stand in a line such that two of them,$A$ and $B$,are always together?

How many numbers consisting of $5$ digits can be formed in which the digits $3, 4$ and $7$ are used only once and the digit $5$ is used twice?

The exponent of $3$ in $100!$ is