Arranging seven charms on a bracelet

Question

Seven distinct charms are placed around a bracelet, forming a closed loop. Two bracelets count as the same if one can be turned into the other by rotating it, and also if one is the mirror image of the other (because a bracelet can be flipped over).

How many distinct bracelets are possible?

Accepted Answer

Start with the circular count, treating only rotations as equivalent: that is (7-1)! = 6! = 720. Now account for flipping. A bracelet can be turned over, so each arrangement and its mirror image (the reverse order around the loop) are the same object. This pairs up the 720 circular arrangements into groups of 2 (an arrangement and its reflection), so we divide by 2: \frac{(7-1)!}{2} = \frac{720}{2} = 360. Takeaway. A bracelet or necklace allows both rotation and reflection, so the count is \dfrac{(n-1)!}{2}. Seven charms give 360.

Arranging seven charms on a bracelet

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