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Math Help - Arranging into pairs

  1. #1
    Member oldguynewstudent's Avatar
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    Arranging into pairs

    Just need to check if I understand this concept:

    In how many different ways can we arrange 2n people into n pairs?

    The equivalence classes for this problem would contain 2^n ways to rearrange the pairs and n! ways to order the pairs, correct?

    That would leave \frac{(2n)!}{(2^{n})(n!)} as the answer.

    I'm having a little trouble grasping the intricacies of partitioning and equivalence classes.
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  2. #2
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    Quote Originally Posted by oldguynewstudent View Post
    In how many different ways can we arrange 2n people into n pairs?
    The equivalence classes for this problem would contain 2^n ways to rearrange the pairs and n! ways to order the pairs, correct?

    That would leave \frac{(2n)!}{(2^{n})(n!)} as the answer.

    I'm having a little trouble grasping the intricacies of partitioning and equivalence classes.
    What exactly is your question?
    These are known as unordered partitions.
    If we have m\cdot n people to divide into n groups of m each, that can be done in
    \frac{(m\cdot n)!}{(m!)^n(n!)} ways.
    Last edited by Plato; June 2nd 2010 at 01:55 PM.
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  3. #3
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Plato View Post
    What exactly is your question?
    These are known as unordered partitions.
    If we have m\cdot n people to divide into n groups of m each, that can be done in
    \frac{(m\cdot n)!}{(m!)^n(n!)} ways.
    Thank you, I just wanted to make sure I understood what I was doing.

    This question is part of a section on partitions \equiv equivalence classes.

    There was no general formula given.
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    Quote Originally Posted by oldguynewstudent View Post
    This question is part of a section on partitions \equiv equivalence classes.
    Just one comment: I donít think this has anything to do with equivalent classes in general.
    How do you understand it?
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  5. #5
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Plato View Post
    Just one comment: I donít think this has anything to do with equivalent classes in general.
    How do you understand it?
    It is easier to show you the context from pages 37 and 38. This is question 42 on page 38.
    Attached Thumbnails Attached Thumbnails Arranging into pairs-comb_37.jpg   Arranging into pairs-comb_38.jpg  
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