In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections : $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$. A student is required to attempt total $15$ questions taking at least $4$ questions from each section. If section $A$ has $8$questions, section $\mathrm{B}$ has $6$ questions and section $\mathrm{C}$ has $6$ questions, then the total number of ways a student can select $15$ questions is

In an examination there are three multiple choice questions and each question has $4 $ choices. Number of ways in which a student can fail to get all answers correct, is

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

If $^{n}{P_4} = 24.{\,^n}{C_5},$ then the value of $n$ is

In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets themselves such that they get consecutive blocks of $5, 3 $ and $2$ tickets is