In how many ways can $mn$ letters be posted in $n$ letter-boxes?

How many $6$-digit numbers can be formed from the digits $0, 1, 3, 5, 7,$ and $9$ which are divisible by $10$,given that no digit is repeated?

The total number of words that can be formed by using all the letters of the word $ALLEN$ such that all vowels occur in alphabetical order is:

Seven scientists $S_1, S_2, \ldots, S_7$ are invited to deliver one lecture each in a conference. The number of ways all the seven lectures can be arranged such that the lecture of $S_1$ is prior to that of $S_3$ and the lecture of $S_3$ is prior to that of $S_7$ is

In how many ways can $5$ examination papers be arranged so that the physics and chemistry papers never come together?

The number of integers greater than a million that can be formed using the digits $2, 3, 0, 3, 4, 2, 3$ is