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Combinatorical Problem

  1. Jan 14, 2004 #1
    Hi all!

    In how many unique ways can 4 dices be combined? Note that the order amongst the dices is not relevant, so 1-2-3-4 = 4-3-2-1.

    My idea is that you select the values, one by one. You can select the first value in 6 ways, the second in 6 ways, the third in 6 ways and the fourth in 6 ways. This yield [tex]6^4[/tex] combinations. The order was irrelevant, so the answer should then be [tex]\frac{6^4}{4!}[/tex].

    This is obviously wrong... I'm trying to figure out how to think to solve a problem like this.

    The answer is

    [tex]{6+4-1} \choose {4}[/tex]

    Which basically means "select 4 of the 6, and put each value back when you've selected it". I don't get this...

    Would really appreciate some guidance!
  2. jcsd
  3. Jan 14, 2004 #2
    Ok, I have an idea.

    Lets say we have dices in a line. The first dice has the value 1, the second 2, the third 3...the sixth 6. This yields the equation

    [tex]x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 4[/tex]

    So the solution [tex]x_1 = 2, x_3 = 1, x_4 = 1[/tex] means that two dices has the value 1, one has 3 and one has 4.

    This equation has the solution
    [tex]{{4 + 6 - 1} \choose {4}} = {{4+6-1} \choose {5}}[/tex]

    Can this be solved in some other way?
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