Joanna has six beads that she wants to assemble into a bracelet. Two of the beads have the same color, and the other four all have different colors. How many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical if one can be rotated and/or reflected to obtain the other.)
We can place one of the unique beads, and call it reference number 1. Then the possible places of the two same colored beads are at 2 and 6, 3 or 5 and 4, so three ways. The rest of the 4 charms can be placed in 4! ways. So the number of arrangements is 3*4! = 72.