2013 AMC 12A Problems/Problem 15
Problem
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. 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?
Solution 1
There are two possibilities regarding the parents.
1) Both are in the same store. In this case, we can treat them both as a single bunny, and they can go in any of the 4 stores. The 3 baby bunnies can go in any of the remaining 3 stores. There are combinations.
2) The two are in different stores. In this case, one can go in any of the 4 stores, and the other can go in any of the 3 remaining stores. The 3 baby bunnies can each go in any of the remaining 2 stores. There are combinations.
Adding up, we get combinations.
Solution 2
We tackle the problem by sorting it by how many stores are involved in the transaction.
1) 2 stores are involved. There are ways to choose which of the stores are involved and 2 ways to choose which store recieves the parents. total arrangements.
2) 3 stores are involved. There are ways to choose which of the stores are involved. We then break the problem down to into two subsections - when the parents and grouped together or sold separately.
Separately:
All children must be in one store. There are ways to arrange this. ways in total.
Together: Both parents are in one store and the 3 children are split between the other two. There are ways to split the children and ways to choose to which store each group will be sold. .
total arrangements.
3) All 4 stores are involved. We break down the problem as previously shown.
Separately:
All children must be split between two stores. There are ways to arrange this. We can then arrange which group is sold to which store in ways. .
Together: Both parents are in one store and the 3 children are each in another store. There are ways to arrange this.
total arrangements.
Final Answer:
.
Video Solution
~savannahsolver
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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All AMC 12 Problems and Solutions |
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