There are $6$ candidates for $3$ posts. In how many ways can the posts be filled?

The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

$^nC_r + ^nC_{r-1}$ is equal to

In how many ways can $5$ boys and $5$ girls stand in a row so that no two girls may be together?

How many $3$-digit numbers are there,with no digits repeated?

There are $10$ different letters of the English alphabet. If words of $5$ letters are formed using these letters,how many words can be formed such that at least one letter is repeated?