Shades of paint
Probability
Level
4
You have two full cans each of red, yellow and blue paint, all the same size. You pour any number of entire cans into a large empty vat and stir. How many possible paint colors could you produce in this manner?
The answer is 19.
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1 solution
You have three choices for how many cans of red paint to use: zero, one or two.
Likewise, you have three choices for each of yellow and blue.
This would yield 3 x 3 x 3 = 27 possibilities.
But this overcounts, in two ways.
We must exclude the (0, 0, 0) configuration as that yields no paint at all!
We've duplicated colors; the color produced by 1 red, 1 yellow is exactly the same as by 2 red, 2 yellow.
In fact, it is exactly the paint configurations that use no ones (only 0s and 2s) that we want to subtract off, because they're either nonsense like (0, 0, 0) or they are an exact doubling of another configuration we counted.
And there are 2 x 2 x 2 = 8 such configurations. 27 - 8 = 19.