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Thread: 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 $\displaystyle {8\choose2}{5\choose 2}=280$

    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: $\displaystyle \frac{9!}{3!3!3!}\cdot\frac{1}{3!}=280$

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

    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 $\displaystyle \frac{9!}{(3!)^3}$ 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 $\displaystyle 3!$.

    If you want to divide twenty people into four groups of five each the number is $\displaystyle \frac{20!}{(5!)^4(4!)}$.
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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 $\displaystyle \frac{9!}{(3!)^3}$ 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 $\displaystyle 3!$.

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

    Is $\displaystyle 3!$ in the denominator meant to get rid of the order:
    {RED, BLUE, GREEN}$\displaystyle ,${RED, GREEN, BLUE}$\displaystyle ,${GREEN, BLUE, RED}$\displaystyle ,${GREEN, RED, BLUE}$\displaystyle ,${BLUE, GREEN, RED}$\displaystyle ,${BLUE, RED, GREEN}=3!
    Last edited by novice; Dec 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 $\displaystyle 3!$ 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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