If ${}^{27}P_{r+7} = 7722 \times {}^{25}P_{r+4}$,then $r = $

How many $5$-digit numbers can be formed using the digits $1, 2, 3, 4, 5$ such that the number is divisible by $4$?

How many arrangements can be made with $5$ boys and $5$ girls in a row such that boys and girls alternate?

The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

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 . . . . . . .

How many numbers lying between $999$ and $10000$ can be formed with the help of the digits $0, 2, 3, 6, 7, 8$ when the digits are not to be repeated?