From among the $36$ teachers in a school,one principal and one vice-principal are to be appointed. In how many ways can this be done?

The number of words that can be formed out of the letters of the word $ARTICLE$ so that the vowels occupy even places is

How many numbers lying between $99$ and $1000$ can be formed using the digits $2, 3, 7, 0, 8, 6$,if each digit can be used only once in each number?

There are $6$ men and $8$ women in a club. $A$ committee of $5$ people has to be formed. What will be the number of ways to select $5$ people if $2$ men and $3$ women are selected?

If $n = ^mC_2,$ then the value of $^nC_2$ is given by

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