Find 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.

Explanation

Here, we have to arrange p things out of n, `1 le p le r`, and repetition is allowed. When p=1, the number of permutations is n. When p=2, the number of permutations is `n xx n=n^(2)`.
(Since repetition is allowed, first thing can be taken in n ways and the second thing can also be taken in n ways.)
When p=3, the number of permutations is `n xx n xx n =n^(3)`. When p=r, the number of permutations is `n xx n xx n .. r`times `=n^(r )`
Hence, the total number of permutations is
`n+n^(2)+n^(3)+.. +n^(r )=(n(n^(r )-1))/((n-1))`