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Math Help - Circular Keychain Problem

  1. #1
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    Circular Keychain Problem

    For exam review, I did this question:

    "How many ways can 6 keys be placed on a circular key ring? Both sides of the ring are the same, and here is no way to tell which is the "first" key on the ring."

    My first thought is P(6,6)/6 = 120. The idea being that each permutation "matches" five other permutations when rotated around the ring. However, the book answer is 60, not 120, so evidently there is something off in my thinking.
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  2. #2
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    Re: Circular Keychain Problem

    you're missing the fact not only can you do 6 shifts that don't affect the circular order, you can also rotate the entire ring 180 degrees so you see the reverse order of all the sequences. This cuts the number down by a further factor of 2 which gets you $\dfrac {6!}{6\cdot 2} = 60$
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  3. #3
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    Re: Circular Keychain Problem

    Hello, infraRed!

    How many ways can 6 keys be placed on a circular key ring?
    Both sides of the ring are the same, and here is no way to tell which is the "first" key on the ring.

    My first thought is P(6,6)/6 = 120. . Good!
    The idea being that each permutation "matches" five other permutations when rotated around the ring.
    However, the book answer is 60, not 120, so evidently there is something off in my thinking.

    I assume that the 6 keys are distinguishable.


    We can think of it this way . . .

    The first key can be placed anywhere on the keyring.
    Then the other 5 keys can be placed in relation to the first key.
    Hence, there are: . 5! = 120 ways.


    But the keyring can be "flipped".

    Code:
                    So that
            A                     A
           / \                   / \
          F   B                 B   F
          |   |   is equal to   |   |
          E   C                 C   E
           \ /                   \ /
            D                     D
    And that is why your answer is twice as large.


    In the future, beware of problems involving keyrings, bracelets, or necklaces.
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