Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is

Let $S$ be the set of all passwords which are $6$ to $8$ characters long,where each character is either an alphabet from $\{A, B, C, D, E\}$ or a number from $\{1, 2, 3, 4, 5\}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from $\{1, 2, 3, 4, 5\}$ is $\alpha \times 5^{6}$,then $\alpha$ is equal to $.......$

Two packs of $52$ cards are shuffled together. The number of ways in which a man can be dealt $26$ cards so that he does not get two cards of the same suit and same denomination is

The number of ways to distribute $30$ identical candies among four children $C_{1}, C_{2}, C_{3}$ and $C_{4}$ such that $C_{2}$ receives at least $4$ and at most $7$ candies,and $C_{3}$ receives at least $2$ and at most $6$ candies,is equal to

The number of $5$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6$ such that the number must include both $1$ and $2$ is:

There are three sections in a question paper,each containing $4$ questions. If a candidate has to answer only $5$ questions from this paper without leaving any section,then the number of ways the candidate can make the choice of questions is: