In how many ways can I place 5 different beads on a bracelet if flipping or rotating the bracelet does not change the order of the beads?

Answer 1

The symmetry group here is D_5, see here:

http://en.wikipedia.org/wiki/Dihedral_group

So, you have 10 different rotations and reflections (the identity, i.e. doing nothing is one of these operations).

If you don't take into account this symmetry, you would have had 5!=120 possibilities. Applying different elements of D_5 always gives you a different configuration (this follows from the fact that you have 5 different beads), so the number of different configurations that can't be obtained from each other by applying a rotation or reflection is 120/10 = 12.

If the beds are not all different, then it can happen that two different elements of D_5 applied to the same configuration yield the same result, dividing by 10 then won't give the correct result. In that case, you have to apply Pólya enumeration theorem:

http://en.wikipedia.org/wiki/P%C3%B3lya%27s_enumeration_theorem

Answer 2

Awesome, thanks!