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Math Help - How many unique objects...

  1. #1
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    How many unique objects...

    Hello,

    This is a permutation or combinatorics problem which I really don't know if falls under pre-university math.

    I have a pool of base shapes (circle, square, etc.) [9]

    I have a pool of base colors (red, blue, etc.) [7]

    I would like to combine them in such a way that each shape gets 1 color (red) or a combination of two (red/blue == blue/red)

    I would like the result count of these combinations.

    Writing it out with smaller numbers produced the formula with these numbers as 9 * (7 + 6 + 5 + 4 + 3 + 2 + 1) = 252.

    Am I doing it right?
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  2. #2
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    Re: How many unique objects...

    deleted as it was incorrect
    Last edited by romsek; February 17th 2014 at 11:21 AM.
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  3. #3
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    Re: How many unique objects...

    Romsek, the problem with your approach is that 7 x 6 combos of two colors means that a conbination of red/blueis considered different from a combination of blue/red, but the OP said they are considered to be equivalent. You have to divide that result by 2 to limit it to unique combinations of two colors. So there are 7x6/2 combinations of 2 colors plus 7 choices of 1 color, for a total of 28 possible color c0mbinations.

    Another way to think about this is to define the case of a sngle color as the combinatio of that color and a null color. Now there are always two colors, one of which may be "null." The number of unique ways to combine these 8 possibilities without repeating the color is 8x7/2 = 28.

    Now multiple the 28 possible color combinations by 9 possible shapes to get the total number of options.
    Last edited by ebaines; February 17th 2014 at 11:12 AM.
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  4. #4
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    Re: How many unique objects...

    Quote Originally Posted by Kcolaid View Post
    This is a permutation or combinatorics problem which I really don't know if falls under pre-university math.
    I have a pool of base shapes (circle, square, etc.) [9]
    I have a pool of base colors (red, blue, etc.) [7]
    I would like to combine them in such a way that each shape gets 1 color (red) or a combination of two (red/blue == blue/red)
    I would like the result count of these combinations.
    I am not really this is a correct reading of your question. It is different from reply #2.

    It seems that you have nine different objects to paint using seven different colors, each object can have one or two colors.
    If that correct then I say that the answer is:


    I do not think that the order in which the colors are chosen matters here. Nor have I considered how the shapes are colored.
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    Re: How many unique objects...

    Quote Originally Posted by Plato;810592
    It seems that you have nine different objects to paint using seven different colors, each object can have one or two colors.
    If that correct then I say that the answer is:
    [img
    http://www.texify.com/img/%5CLARGE%5C%219%20%5Ccdot%20%5Cleft%28%5Cbinom%7B7 %7D%7B2%7D%2B%5Cbinom%7B7%7D%7B1%7D%5Cright%29%3D2 52.gif[/img]

    I do not think that the order in which the colors are chosen matters here. Nor have I considered how the shapes are colored.
    That is correct. How the shapes are colored is irrelevant and that they are shapes important. Just wanted something relate-able. The answer I got is the same as the one you provided so I can only assume it's right even if I wrote it differently.
    (Don't know what (7 2) (7 1) signify)
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  6. #6
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    Re: How many unique objects...

    $$\left(\begin{array}{c}&7 \\ &2 \end{array}\right)$$

    is "7 choose 2", i.e. the binomial coefficient

    it's given by

    $$\left(\begin{array}{c}&n \\ &k \end{array}\right) = \frac{n!}{k!(n-k)!}$$
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  7. #7
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    Re: How many unique objects...

    Quote Originally Posted by romsek;810627

    is "7 choose 2", i.e. the [URL="http://en.wikipedia.org/wiki/Binomial_coefficient"
    binomial coefficient[/URL]
    Thanks! Really appreciate the info.
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