Question

Find the total number of permutation of $k$ different things in a row taken not more than $r$ at a time (each thing may be repeated any number of times )

Answer 1

When we arrange one thing at a time, the number of possible permutations is $k$

When we arrange them two at a time the number of possible permutations are $k\times k=k^2$ and so on.Thus the total number of permutations are

$k+k^2+k^3+...+k^r=\frac{k(k^r-1)}{k-1}$ for $n>1$