$P, Q, R$ and $S$ have to give lectures to an audience. The organiser can arrange the order of their presentation in ............. ways.

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken $3$ at a time without repetition and each arrangement containing at least one vowel,is

The total number of $3$-digit numbers,whose sum of digits is $10$,is

In an election,there are $5$ candidates and $3$ vacancies. $A$ voter can vote for a maximum of $3$ candidates. In how many ways can a voter cast their vote?

Everybody in a room shakes hands with everybody else. The total number of handshakes is $66$. The total number of persons in the room is:

How many $4$-digit numbers are there with distinct digits?