If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$,then the value of $r$ is equal to:

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

The number of three-digit numbers $\overline{abc}$ such that the arithmetic mean of $b$ and $c$ is equal to the square of their geometric mean is

The number of $5$ digit telephone numbers having at least one of their digits repeated is

There are three sections in a question paper,each section containing $4$ questions. If a candidate has to answer exactly $5$ questions from this paper such that at least one question is answered from each section,then the number of ways in which a candidate can make the choice of questions is

If $_n{P_4} = 720 \binom{n}{r}$,then $r = ..........$