Problem:
Zara has a collection of marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let and represent the Steelie and the Tiger in some order, with always coming before . To place and , we can have the following cases:
This leaves configurations, with the underscores being the reserved spots for the other marbles. With any other configuration, we would have and next to each other or before .
Now, each configuration can have either the Steelie be and the Tiger be , or the Steelie be and the Tiger be . This doubles the number of configurations, making the total number of configurations .
Also, each configuration can have either the Bumblebee be the first spot and Aggie be the second spot, or vice versa. This again doubles the number of configurations, making the total number of configurations .
Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions