The number of $6$-digit numbers that can be formed using the digits $0, 1, 2, 5, 7,$ and $9$ which are divisible by $11$,where no digit is repeated,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?

For a set of $5$ true or false questions,no student has written all the correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

All five-letter words are made using all the letters $A, B, C, D, E$ and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $W_{n}$. Let the probability $P(W_{n})$ of choosing the word $W_{n}$ satisfy $P(W_{n}) = 2P(W_{n-1}), n > 1$. If $P(CDBEA) = \frac{2^{\alpha}}{2^{\beta}-1}, \alpha, \beta \in N$,then $\alpha + \beta$ is equal to

There are $3$ sections in a question paper and each section contains $5$ questions. $A$ candidate has to answer a total of $5$ questions,choosing at least one question from each section. Then the number of ways,in which the candidate can choose the questions,is

From a collection of $20$ consecutive natural numbers, four are selected such that they are not consecutive. The number of such selections is