2014 AMC 12A 第 3 题

Question

Ralph 沿 Jane 街走，依次经过四栋房子，每栋房子颜色不同。他先经过橙色房子，再经过红色房子；并且先经过蓝色房子，再经过黄色房子。蓝色房子不与黄色房子相邻。彩色房子的排列顺序有多少种可能？

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

2

3

4

5

6

答案：B

知识点：有限制的排列分类讨论

难度评级：1200

小提示：

按橙色房子先出现还是蓝色房子先出现分类。

Split into cases by whether the orange house or the blue house comes first

大提示：

因为蓝色和黄色不能相邻，所以它们之间至少隔一栋房子。

Because blue and yellow cannot be adjacent, at least one house must separate them

解答：

若橙色先出现，则蓝色和黄色不能相邻，迫使顺序为橙、蓝、红、黄。

若蓝色先出现，则黄色可以在第三或第四个位置（不能在第二个位置以避免相邻），得到蓝、橙、黄、红和蓝、橙、红、黄。

这就是全部

3

种有效顺序。

所以正确答案是 B。

If orange comes first, then blue and yellow cannot be adjacent, forcing the order orange, blue, red, yellow.

If blue comes first, yellow can be in the third or fourth position (never second, to avoid adjacency), giving blue, orange, yellow, red and blue, orange, red, yellow.

These are the only

3

valid orderings.

Thus, the correct answer is B.

2014 AMC 12A 第 3 题

答案：B

难度评级：1200

小提示：

大提示：

解答：

其他年份的第 3 题