A circular necklace contains n beads. Each bead is black or white. How many different necklaces can be made with n beads?

There is no clasp to identify a specific point on the chain, and a flipped over necklace is still the same necklace.
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To get you started:

With 1 bead, the necklace can be either 1 black or 1 white bead.

With 2 beads, the necklace can be either 2 black, 2 white, or 1 black-1 white

With 3 beads, the necklace can be either 3 black, 3 white, 2 black-1 white, 2 white-1 black, etc...

# Beads  Number of Necklaces
   1          2
   2          3
   3          4
   4          6
   5          8
   6         13

This looks to me like a permutation problem rather than a combination problem, since it is a different necklace depending upon the order of the beads. For each bead, their are two possible choices, black or white, and we would multiply by two for each bead on the necklace (n). Therefore, the formula is 2 to the n power.

Posted by Candy on 2004-08-22 09:01:50