If $0 < r < s < n$ and ${}^n P_r = {}^n P_s$,then find the value of $r + s$.

All possible numbers are formed using the digits $1, 1, 2, 2, 2, 2, 3, 4, 4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is

The number of ways in which $5$ girls and $3$ boys can be seated in a row so that no two boys are together is:

How many words,with or without meaning,each of $3$ vowels and $2$ consonants can be formed from the letters of the word $INVOLUTE$?

The number of words that can be formed by using the letters of the word $CALCULATE$ such that each word starts and ends with a consonant,are

$6$ 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 $p/q =$