In how many different ways can the letters of the word '$PRAISE$' be arranged?

How many $3$-digit even numbers can be formed using the digits $0, 2, 3, 5, 7$ when repetition is not allowed?

In a party,$12$ persons are to be arranged around a round table. If two particular persons are not to be seated side by side,then the total number of arrangements is

$5$-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

The number of values of $r$ satisfying $^{69}C_{3r-1} - ^{69}C_{r^2} = ^{69}C_{r^2-1} - ^{69}C_{3r}$ is:

The number of $7$-digit numbers which can be formed using the digits $1, 2, 3, 2, 3, 3, 4$ is: