How many ways can we choose a pair of edges of a dodecahedron, up to rotational symmetry?
Taking $180$ rotation we have $15$ axis of rotations.
Since there are $6$ pairs of opposite faces, we have an axis going through every pair of opposite faces and have 4 non-trivial rotations for every pair of opposite faces. There are $4*6 = 24$ such rotations.
Lastly, at the opposite pairs of vertices and we may rotate the axis which passes through both vertices and have $2$ more rotations for each pair of vertices. Here we have $2*10 = 20$ such rotations.
Finally we have the identity.
Hence there are $15+24+20+1 = 60$ rotational symmetries of a dodecahedron.
After this I am not able to proceed. Need some hints.
Source: Homework.