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2013 AMC 12A Problem 15

Problem 15 of 25IntermediateCounting & Probability

Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cottontail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?

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Solution

If the two parents share a store, there are 44 choices for it, and each child must go to one of the other three stores: 4⋅33=1084\cdot 3^3 = 108 ways. If the parents go to different stores, there are 4⋅3=124\cdot 3 = 12 choices, and each child must go to one of the two remaining stores: 12⋅23=9612\cdot 2^3 = 96 ways. The total is 108+96=204.108 + 96 = 204. Thus, the correct answer is D.

More practice

Concepts: casework · multiplication principle

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.