2012 AMC 12B 第 16 题

Question

Amy、Beth 和 Jo 听了四首不同的歌，并讨论她们喜欢哪些歌。没有一首歌被三人都喜欢。此外，对这三个女生中的每一对，都至少有一首歌被这两人喜欢、但不被第三人喜欢。有多少种不同的可能情况？

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

答案：B

知识点：分类讨论乘法原理

难度评级：1840

小提示：

每首歌要么被

\{AB\},\{AC\},\{BC\}

中的一对喜欢，要么只被一个女生喜欢，要么无人喜欢；三对都必须出现

Each song is liked by one of the pairs

\{AB\},\{AC\},\{BC\},

by a single girl, or by no one; all three pairs must appear

大提示：

分成两种情况：四首歌都分给成对喜欢的组合，或者恰好三首歌覆盖三对组合

Split into the case where all four songs go to pairs and the case where exactly three songs cover the three pairs

解答：

每首歌恰好被三对中的一对喜欢，或被单个女生喜欢，或无人喜欢。每一对都必须被表示出来。

情况 $1$ ： 每首歌都被一对女生喜欢。某一对得到四首歌中的两首（

\binom42=6

种，且有

3

种选择是哪一对），另外两对各得到一首歌（

2

种）。共有

3\cdot6\cdot2=36

。

情况 $2$ ： 三首歌分别对应三对女生（各一首），第四首歌被单个女生喜欢或无人喜欢。把四首歌分配到这四个角色有

4!=24

种，剩余角色有

4

种选择（Amy、Beth、Jo 或无人）：

24\cdot4=96

。

总数为

36+96=132

。

因此正确答案是 B。

Each song is liked by exactly one of the three pairs, by a single girl, or by no one. Every pair must be represented.

Case $1:$ every song is liked by a pair. One pair gets two of the four songs (

\binom42=6

ways, and

3

choices for which pair), and the other two pairs get one song each (

2

ways). This gives

3\cdot6\cdot2=36.

Case $2:$ three songs go to the three pairs (one each) and the fourth song is liked by a single girl or no one. Assigning the four songs to these four roles gives

4!=24

ways, and the leftover role has

4

options (Amy, Beth, Jo, or no one):

24\cdot4=96.

The total is

36+96=132.

Thus, the correct answer is B.

2012 AMC 12B 第 16 题

答案：B

难度评级：1840

小提示：

大提示：

解答：

其他年份的第 16 题