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Math Help - Necklace

  1. #1
    Senior Member pankaj's Avatar
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    Necklace

    Find number of ways to make a necklace using 3 identical diamonds and 15 identical pearls.
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  2. #2
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    Re: Necklace

    This is a "necklace" so circular so we can take anything to be the "starting point" so lets take one of the pearls as marking the "start". That leaves 17 items to be put on. There are 17! ways to order 17 distinct objects. But there are 3 identical diamonds and there are 3! ways to interchange them that we do not want to count. There are 15 identical pearls so 15! ways to interchange them. That gives a total of 17!/(3! 15!) different necklaces.
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  3. #3
    Senior Member pankaj's Avatar
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    Re: Necklace

    \frac{17!}{3! 15!}=\frac{17\times 8}{3}=\frac{136}{3}=Number of necklaces
    Last edited by pankaj; January 22nd 2014 at 07:27 AM.
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  4. #4
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    Re: Necklace

    Quote Originally Posted by pankaj View Post
    \frac{17!}{3! 15!}=\frac{17\times 8}{3}=\frac{176}{3}=Number of necklaces
    However, the whole question is much more complicated then that.

    You need to study this webpage.
    You will see the you must account for reflections and rotations.

    If you calculate \frac{17!}{3! 15!}=\frac{17\times 8}{3}=\frac{176}{3} you will see that is not a whole number.
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