Counting the necklaces

Statement

Using beads of $3$ colors we form strings of exactly $5$ beads; there are $3^5$ possible strings. We discard the monochromatic strings and join the rest into loops: rotating a string yields $5$ strings giving the same necklace. How many distinct (non-monochromatic) necklaces can be formed?

Solution

Total strings: $3^5=243$. The monochromatic ones number $3$ (one per color), leaving $243-3=240$ non-monochromatic strings. Since $5$ is prime, each necklace corresponds to exactly $5$ strings (its rotations), all distinct. Hence $N=\frac{3^5-3}{5}=\frac{240}{5}=\boxed{48}.$ (This is the combinatorial idea behind Fermat’s little theorem: $5\mid 3^5-3$.)