Bad Cutting
A father has 3 identical cupcakes to distribute to his triplets. But, due to a blackout, his triplets directly helped themselves to the cupcakes, and there might be some of them who didn't get 1.
How many different ways could the triplets help themselves to the cupcakes, so that there wasn't an even distribution?
The answer is 9.
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1 solution
This problem should have specified that the pieces are indistinguishable. Suppose the "3 equal pieces" have colors red, green, blue. Then there would be 27 different ways of distributing the pieces, and 6 ways to do it right, or 21 ways to do it wrong.
If the pieces are indistinguishable, then either one of the triplets gets all 3, or one of the triplets gets 2, while one of the remaining gets 1. The number of ways of getting it wrong is 3 + 3(2) = 9.
In quantum physics, the distinction between distinguishability and indistinguishability plays a crucial role, as for example, if the pieces of the cake are individual electrons, then there can only be a total of 10 ways to distribute them, not 27. This can lead to unexpected statistical results.