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Determine the number of necklaces of 8 beads each of n colors. ( I know we...

Determine the number of necklaces of 8 beads each of n colors. ( I know we use Burnside's theorem, but I dont know how to apply it)

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Answer #1

There are 8 beads in that necklace and each one is a different color.

Identify the colors by the codes 1, 2, 3, 4, 5, 6, 7 and 8

Lay the necklace on the table and look for the bead of color 8

What are the other colors read clockwise from that bead?

Is it 1234567? 1532476? Some other sequence

There are 7! = 5040 such sequences.

However, 1234567 and 7654321 represent the same necklace flipped over.

By laying the necklace on the table we made it easier to count the possibilities, but we counted each one twice.

There are 5040/2 = 2520 possible necklaces

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