# Thread: How many unique objects...

1. ## 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?

2. ## Re: How many unique objects...

deleted as it was incorrect

3. ## 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.

4. ## Re: How many unique objects...

Originally Posted by Kcolaid
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.

5. ## Re: How many unique objects...

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)

6. ## 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

$$\left(\begin{array}{c}&n \\ &k \end{array}\right) = \frac{n!}{k!(n-k)!}$$

7. ## Re: How many unique objects...

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.