My friend lost 2 charms off her 7-charm bracelet. For her birthday, I bought her a new charm to replace one of the lost ones. Unfortunately, I messed up and got her a duplicate of one of the charms she still has. How many distinguishable ways can she put her 6 charms on her bracelet? (Two of the charms are the same, rotations are indistinguishable, and turning the bracelet front-to-back is indistinguishable.)

Question

anonymous · Answer

hii

anonymous · Answer

place one of the unique charms and make it the reference number, 1 
the possible distinct placements (in a circle) are 2 or 6, 3 or 5 and 4, ie 3 ways 
the remaining 4 unique charms can be placed in 4! ways 
thus total number of ways = 3*4 = 72