Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty?

The numbers $1, 2, 3, \ldots, n$ are arranged in a random order. The probability that the digits $1, 2, 3, \ldots, k$ appear as a block in that order is:

Let $n$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue such that all the girls stand consecutively. Let $m$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue such that exactly four girls stand consecutively. Then the value of $\frac{m}{n}$ is

From a group of $7$ batsmen and $6$ bowlers,$10$ players are to be chosen for a team,which should include at least $4$ batsmen and at least $4$ bowlers. One batsman and one bowler,who are the captain and vice-captain of the team respectively,must be included. The total number of ways such a selection can be made is:

The number of all possible positive integral solutions of the equation $xyz=30$ is

The number of natural numbers,between $212$ and $999$,such that the sum of their digits is $15$,is . . . . . .