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Math Help - Binomial expansion + interesting question

  1. #1
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    Binomial expansion + interesting question

    Hey,

    how would I use a 'combinatoral proof' to prove this:


    and here's a question that has got me puzzled..
    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.
    (a) If the pairing is done at random, what is the probability that there are no offensive-defensive pairs?
    (b) What is the probability that there are 2i offensive-defensive roommate pairs (with i= 1,2,...,10)

    Thanks in advance
    Last edited by fatlucky; February 1st 2011 at 02:14 PM.
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  2. #2
    MHF Contributor harish21's Avatar
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    a)
    total of 40 players are grouped into two..

    players can be sorted into ordered groups in \dbinom{40}{2,2,....,2}=\dfrac{40!}{(2!)^{20}} ways.

    now find the number of ways players can be sorted into unorderd pairs ......(I)

    ------------------------------------------------------------------------------------------------

    also, a division will give you no offensive-defensive pairs if the offensive (and therefore the defensive) players are paired among themselves. So there are \dfrac{20!}{10! \cdot 2^{10}} divisions for offensive as well as defensive players...(II)

    ------------------------------------------------------------------------------------------------

    Using (I) and (II), you can now find the probability that there are no offensive-defensive pairs!
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  3. #3
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    Quote Originally Posted by fatlucky View Post
    Hey,

    how would I use a 'combinatoral proof' to prove this:


    [snip]
    Google
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