The number of ordered pairs ( $\mathrm{r}, \mathrm{k}$ ) for which $6 \cdot ^{35} \mathrm{C}_{\mathrm{r}}=\left(\mathrm{k}^{2}-3\right)\cdot{^{36} \mathrm{C}_{\mathrm{r}+1}}$. where $\mathrm{k}$ is an integer, is

A car will hold $2$ in the front seat and $1$ in the rear seat. If among $6$ persons $2$ can drive, then the number of ways in which the car can be filled is

$^{47}{C_4} + \mathop \sum \limits_{r = 1}^5 {}^{52 - r}{C_3} = $

In an election there are $8$ candidates, out of which $5$ are to be choosen. If a voter may vote for any number of candidates but not greater than the number to be choosen, then in how many ways can a voter vote

The total number of ways of selecting six coins out of $20$ one rupee coins, $10$ fifty paise coins and $7$ twenty five paise coins is

To fill $12$ vacancies there are $25$ candidates of which five are from scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all, then the number of ways in which the selection can be made