Question

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?

$108$

$132$

$671$

$846$

$1105$

Solution:

There are two cases: Each pair has exactly one liked song in common, or some pair has $2$ liked songs in common. This is because each pair must have at least $1$ liked song in common, and any more pairs than in the cases would result in $5$ songs.

Case 1: Each pair has exactly one liked song in common

There are $4$ ways to choose the song that one pair likes, $3$ ways to choose the song that the second pair likes, and $2$ ways to choose the song the third pair likes if we choose some order for them. Then, for the last song, one of them could like it which has $3$ cases or none of them likes it which is another case. Thus, the number of solutions in this case is $4\cdot 3\cdot 2\cdot (3+1)=96.$

Case 2: Some pair has $2$ liked songs in common

There are $3$ ways to choose the pair that has $2$ liked songs in common. Then, there are $\binom 43 = 6$ ways to choose which songs they like. Finally, there are $\binom 21 =2$ ways to figure out who likes the last song. Thus, the number of solutions in this case is $3\cdot 6\cdot 2=36$

The total amount is then $96+36 = 132.$

Thus, the correct answer is B .

2012 AMC 10B Problem 24

Solution:

Problem 24 in Other Years