Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 6E / b

For the rotation group $C_5$, the identity fixes all $n^5$ colorings. Because five is prime, each of the four nonidentity rotations is a single five-cycle and fixes exactly the $n$ constant colorings. Burnside lemma therefore gives the number of coloring necklaces

For the dihedral group $D_{10}$, the five reflections each fix one vertex and interchange two pairs of vertices, so each fixes $n^3$ colorings. Adding these to the identity and rotation contributions gives the number of coloring bracelets