$6$ 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 $p/q =$

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

Evaluate $4! - 3!$

Find the number of $4$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5$ if no digit is repeated. How many of these will be even?

Among the $4$-digit numbers formed using the digits $0, 1, 2, 3, 4$ where repetition of digits is allowed,the number of numbers which are divisible by $4$ is:

How many words can be formed by arranging the letters of the word $MISSISSIPPI$ such that no two $S$ are together?