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?

Let $x$ denote the number of ways of arranging $m$ boys and $m$ girls in a row so that no two boys sit together. If $y$ and $z$ denote the number of ways of arranging $m$ boys and $m$ girls in a row and around a circular table respectively so that boys and girls sit alternately,then $x: y: z=$

The number of natural numbers less than $10000$ which are divisible by $5$ and in which no digit is repeated is:

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$,if all digits are to appear in the same number at least once,is

If $_n{P_4} = 24 \times \binom{n}{5}$,then $n = \dots$

If $^nP_3 + ^nC_{n-2} = 14n$,then $n = $