It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?

In how many ways can the letters of the word $PERMUTATIONS$ be arranged if the vowels are all together?

In how many ways can the letters of the word $GARDEN$ be arranged such that the vowels appear in alphabetical order?

The number of natural numbers less than $1000$ in which no digit is repeated is

Compute $\frac{7!}{5!}$

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 . . . . . . .