Find the number of $4$ letter words,with or without meaning,which can be formed out of the letters of the word $ROSE$,where the repetition of the letters is not allowed.

$5$-digit numbers are to be formed using $2, 3, 5, 7, 9$ without repeating the digits. If $p$ is the number of such numbers that exceed $20000$ and $q$ is the number of those that lie between $30000$ and $90000$,then $p : q$ is

In a library,there are $a$ books of type $A$,$2$ books of type $B$,$3$ books of type $C$,and $1$ book of type $D$. In how many ways can these books be arranged?

Find $r$ if $^{5}P_{r} = ^{6}P_{r-1}$.

If four-digit numbers are formed by using the digits $1, 2, 3, 4, 5, 6, 7$ without repetition,then how many of these numbers are exactly divisible by $25$?

Evaluate $7!$