The number of ways in which $10$ persons can go in two boats so that there may be $5 $ on each boat, supposing that two particular persons will not go in the same boat is

A person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$, then $n$ equals

Total number of $6-$digit numbers in which only and all the five digits $1,3,5,7$ and $9$ appear, is 

The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is $...........$.

In an election there are $5$ candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote

A test consists of $6$ multiple choice questions, each having $4$ alternative ans wers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is