How many three-digit numbers can be formed from the digits $2, 3, 5, 6, 7,$ and $9$ which are divisible by $5$,given that no digit is repeated?

The number of positive integral solutions of the equation $xyz = 90$ is equal to:

How many numbers of $6$ digits can be formed from the digits of the number $112233$?

If $^{2n}C_3 : ^nC_2 = 44:3$,then for which of the following values of $r$,the value of $^nC_r$ will be $15$?

In how many ways can $6$ letters be posted in $5$ letter boxes available in the locality?

Consider three boxes,each containing $10$ balls labelled $1, 2, dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is: