The total number of permutations of $n (> 1)$ different things taken not more than $r$ at a time,when each thing may be repeated any number of times is

In how many ways can $5$ speakers $S_1, S_2, S_3, S_4$,and $S_5$ give speeches one after the other if $S_3$ must speak after both $S_1$ and $S_2$?

$A$ number lock has $3$ rings and each ring has $8$ digits. The total number of different ways in which the $3$ rings can be rotated is:

The numbers $1, 2, 3,$ and $4$ are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. $A$ person draws two slips from the box,one after the other,without replacement. Describe the sample space for the experiment.

If $^n{P_5} = 9 \times {^{n - 1}}{P_4}$,then the value of $n$ is

Evaluate $8!$.