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

Total number of things = n

repetition of things is allowed and atmost r can be taken at a time. The number of ways of taking 1 thing at a time = n

the number of ways of taking 2 things at a time $=n^2$ the number of ways of taking r things at a time $=n^r$ total permutations $=n+n^2+\mathrm{.......}+n^2$

$=\frac{n\left(n^r-1\right)}{n-1}$