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Math Help - Combinatronics Problem...

  1. #1
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    Exclamation Combinatronics Problem...

    "A football team consists of 20 offensive and 20 defensive players. The players are to be paired in groups of 2 for the purpose of determining roommates. If the pairing is done at random what is the probability that there are no offensive-defensive pairings?"

    My first thoughts were that there are (40Cr2)=720 possible pairings although i'm told its much larger than this...

    Any ideas? Any prods in the right direction would be much appreciated.
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  2. #2
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    Quote Originally Posted by sebjory View Post
    "A football team consists of 20 offensive and 20 defensive players. The players are to be paired in groups of 2 for the purpose of determining roommates. If the pairing is done at random what is the probability that there are no offensive-defensive pairings?"
    The are \frac{40!}{(2^{20})(20!)} random pairings.
    Can you finish?
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  3. #3
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    using patterns i would imagine the amount of OOs and DDs is

    (20!)/((2^10))(10!)) and that the probability would be this divided by your number.

    However could you explain how you arrived at this expression?

    Many thanks

    (p.s. is there a tutorial anywhere on how to use the Math Input function on this forum... the thing denoted by the capital Sigma)
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  4. #4
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    Here is a very quick and dirty way to explain this.
    Say we have a class of 20 freshmen.
    If we divide them up into four study groups: biology, mathematics, literature, and history.

    That can be done in \binom{20}{5}\binom{15}{5} \binom{10}{5} \binom{5}{5} =\frac{20!}{(5!)^4} ways. (  \binom{N}{k} N choose k.)
    Those are known as ordered partitions. Because it makes a difference to a student which group he/she is in.

    Think a class of 20 freshmen is calculus.
    If we divide them up into four groups to review a just returned test.
    Those are known as unordered partitions because only content matters.
    This can be done \frac{20!}{(5!)^4(4!)} ways.
    The details are a bit much to into.
    But I hope you can begin to get the idea.

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    Last edited by Plato; April 10th 2009 at 03:27 PM.
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