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Math Help - Ordered Partition

  1. #1
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    Ordered Partition

    I have this math problem that drives me absolutely nuts. I saw this same one at high school and again in college. Here it is:

    In how many ways can 9 students be divided into three teams?

    The answer to this question is {8\choose2}{5\choose 2}=280

    At high school, I asked my teacher why the answer being it is. He asked me to consider myself a coach. By so, I appoint a student for a team captain for each team and have each team captain choose two others for her team. When I said, "If I were the coach, why can't I choose them myself?" He was speechless.

    Now I see this same problem again in a different book that gave the answer in this form: \frac{9!}{3!3!3!}\cdot\frac{1}{3!}=280

    I understand this part: \frac{9!}{3!3!3!}, but I don't understand why it's multiplied by \frac{1}{3!}

    Can anyone explain?
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  2. #2
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    If you were to gave a RED team, a BLUE team and a GREEN team then \frac{9!}{(3!)^3} would be correct.
    That is because the color team can make a difference. I donít want to be on the BLUE team.

    But because there is nothing to distinguish the teams we divide by another 3!.

    If you want to divide twenty people into four groups of five each the number is \frac{20!}{(5!)^4(4!)}.
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  3. #3
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    Quote Originally Posted by Plato View Post
    If you were to gave a RED team, a BLUE team and a GREEN team then \frac{9!}{(3!)^3} would be correct.
    That is because the color team can make a difference. I don’t want to be on the BLUE team.

    But because there is nothing to distinguish the teams we divide by another 3!.

    If you want to divide twenty people into four groups of five each the number is \frac{20!}{(5!)^4(4!)}.

    Is 3! in the denominator meant to get rid of the order:
    {RED, BLUE, GREEN} ,{RED, GREEN, BLUE} ,{GREEN, BLUE, RED} ,{GREEN, RED, BLUE} ,{BLUE, GREEN, RED} ,{BLUE, RED, GREEN}=3!
    Last edited by novice; December 31st 2009 at 03:06 PM. Reason: replace \cap with ,
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  4. #4
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    Quote Originally Posted by novice View Post
    Is 3! in the denominator meant to get rid of the order:
    Yes, in general.
    Study the other example I gave you.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Yes, in general.
    Study the other example I gave you.
    Thank you, Plato,
    Now I don't have to get up in the middle of the night banging my head.
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