2013 AMC 12A Problem 15

Question

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?

96

108

156

204

372

Answer: D

Concepts:casework multiplication principle

Difficulty rating: 1880

Small hint:

Split into cases by whether the two parents go to the same store

Big hint:

Same store:

4

parent choices and

3^3

child assignments; different stores:

4\cdot 3

and

2^3

Solution:

If the two parents share a store, there are

4

choices for it, and each child must go to one of the other three stores:

4\cdot 3^3 = 108

ways.

If the parents go to different stores, there are

4\cdot 3 = 12

choices, and each child must go to one of the two remaining stores:

12\cdot 2^3 = 96