An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1$-15$ June $2021$ 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

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these

cards are of the same colour?

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

The students $S _{1}, S _{2}, \ldots \ldots, S _{10}$ are to be divided into $3$ groups $A , B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is ........ .

In an election the number of candidates is $1$ greater than the persons to be elected. If a voter can vote in $254$ ways, then the number of candidates is