In how many ways $n$ books can be arranged in a row so that two specified books are not together?

How many $3$-digit numbers can be formed from the digits $1, 2, 3, 4,$ and $5$ assuming that repetition of the digits is not allowed?

Three persons enter a lift at the ground floor. The lift goes up to the $10^{\text{th}}$ floor. If the lift does not stop at the $1^{\text{st}}$, $2^{\text{nd}}$, and $3^{\text{rd}}$ floors, the number of ways in which the three persons can exit the lift at three different floors is equal to . . . . . . .

The range of the function $f(x) = {}^{7 - x}P_{x - 3}$ is

How many $6$-digit numbers can be formed using only odd digits,such that all odd digits appear at least once?

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 $25$?