The number of all four-digit numbers which begin with $4$ and end with either $0$ or $5$ is:

The sum of the digits in the unit place of all numbers formed with the help of $3, 4, 5, 6$ taken all at a time is

Evaluate $5!$

The domain of the function $f(x) = {}^{7-x}P_{x-1}$ is:

If ${}^{27}P_{r+7} = 7722 \times {}^{25}P_{r+4}$,then $r = $

$6$ boys and $5$ girls sit in a line such that $(I)$ no two girls sit together $(II)$ all the girls sit together. If $p$ is the number of arrangements in case $(I)$ and $q$ is the number of arrangements in case $(II)$,then $p/q =$