This problem is the result of many hours of cleaning baby bottles for two daughters. Suppose your baby uses a certain type of bottle that is composed of three pieces (say, top, middle and bottom). The bottles come in two colors (say, blue and green). You have three blue bottles and two green bottles, for a total of five bottles. Your baby uses all five bottles in a single day. Because you’re perpetually tired, you do not disassemble and wash the bottles until the end of the day. After the bottles have dried overnight, you reassemble them and begin your next day. Such is the life of a new parent.
Amanda Montañez
When you go to reassemble the bottles, you notice that you can exchange parts among them so that some are “mixed up.” For example, you can put a green top on a blue middle and blue bottom. Naturally, you try your hand at swapping parts so there is no complete bottle of a single color—in other words, you find a “derangement” of the bottles.
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Here is one such derangement:
Amanda Montañez
Suppose you then buy some new bottles in a third color (say, pink). What is the minimum number of pink bottles required to ensure that no derangement can occur?
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