Problem:
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
Answer Choices:
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B.
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D.
E.
Solution:
There are two cases to consider.
Case 1
Each song is liked by two of the girls. Then one of the three pairs of girls likes one of the six possible pairs of songs, one of the remaining pairs of girls likes one of the remaining two songs, and the last pair of girls likes the last song. This case can occur in ways.
Case 2
Three songs are each liked by a different pair of girls, and the fourth song is liked by at most one girl. There are ways to assign the songs to these four categories, and the last song can be liked by Amy, Beth, Jo, or no one. This case can occur in ways.
The total number of possibilities is .
The problems on this page are the property of the MAA's American Mathematics Competitions