Suppose you are making a necklace and you have n different color beads. How many

Question

Suppose you are making a necklace and you have n different color beads. How many different necklaces are there? Note that if n = 3, a necklace having a red bead, then a blue bead, then a black bead is identical to a necklace having a black bead, then a red bead, then a blue bead (they are equivalent under rotation). Further, it is identical to a necklace having a blue bead, then a red bead, then a black bead (they are equivalent under reflection). (Hint: first count the number of different ways to order the beads. You’ll then have to divide to account for rotation and reflection.)

Explanation / Answer

Counting all permutations of n beads, we get a total of n!. Since we counted each permutation n times due to rotation, we have to divide n!/n and since we counted each of these twice because you can flip the necklace (reflection) over, we have to further divide the total number of permutations by 2. Thus the answer is n!/2n.

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