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Combination Problem

  1. Sep 6, 2011 #1
    1. The problem statement, all variables and given/known data

    You are given 8 balls, each of a different color. How many distinguishable ways can you:

    (1) Divide them (equally or unequally) between 2 urns.
    (2) Divide them (equally or unequally) between 2 children (and each child cares about the colors he or she receives).

    2. Relevant equations

    These are the enumeration formulas we are responsible to know:

    Sampling with replacement and order: [itex]n^r[/itex]
    Sampling without replacement, without order: nCr = [itex]\frac{n!}{r!(n-r)!}[/itex]
    Sampling without replacement, with order: nPr = [itex]\frac{n!}{(n-r)!}[/itex]

    3. The attempt at a solution

    I initially thought that problem (1) would be without replacement and without order, so that the answer would be a combination with n=8 and r=2, and that problem (2) would be without replacement and with order, so a permutation with n=8 and r=2.

    However, that isn't correct. It seems like it might actually be a case where there is replacement. The fact that we are giving the balls to two people, or placing them in two urns, is screwing me up. How can I think about this problem and go about solving it? Is it solvable with just the equations I've listed above? Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 6, 2011 #2

    LCKurtz

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    Think about the 8 balls being lined up in a row.

    _ _ _ _ _ _ _ _

    Put a 1 in each place if the ball goes in urn A and a 0 if it goes in urn B. How many binary numbers does that give? Then it matters whether the urns are distinguishable.
     
  4. Sep 6, 2011 #3
    Okay, so if for each ball there would be 2 options for urns, meaning that for the case when the urns are indistinguishable the options available would be:

    [itex]2^8 = 256[/itex].

    This would be a case where order matters, right? When order doesn't matter I'd have to divide by the number or repetitions, but I'm not understanding how to do that...
     
  5. Sep 6, 2011 #4

    LCKurtz

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    You mean distinguishable
    If you can't tell the urns apart that would just cut it down by half. For example if you put all the balls in urn A and none in urn B, you couldn't distinguish that from its opposite case because you don't know which urn is which.

    I'm afraid the wording of the problem is a bit ambiguous regarding the difference between 1 and 2. The balls are all different colors. If in 1 you are supposed to ignore the colors I would think the problem would state that. In any case we have answered 2, assuming the two children aren't identical twins.
     
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