Math CentralQuandaries & Queries


Question from Sabrina:

How many 2-color (pair) combinations can be made from 12 individual colors... is it (11x12) or (optional): (11x12)x2?

Hi Sabrina,

The answer is dependent on whether order is important. For example suppose you are going to redecorate your living room and you want to choose a color scheme. If you are going to first choose the color of the walls and then the color of the rug then there are $12 \times 11$ choices. You can think of it this way. First you choose the color of the walls and there are 12 possible choices. Regardless of what color you choose you now have 11 possible choices for the rug and hence you have $12 \times 11$ possible choices.

If order is not important and you are just going to choose 2 colors then each pair of colors appears twice in the choices above. For example if ivory and light green are two of the colors then, in the possibilities above, ivory walls and a light green rug is different from light green walls and an ivory rug. Hence if you are going to choose a two color combination and the order you choose them is unimportant then there are $\large \frac{12 \times 11}{2}$ possible combinations.

I hope this helps,

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