Question

Ralph 沿 Jane 街走过连续四栋房子，每栋房子颜色不同。他先经过橙色房子，再经过红色房子；他先经过蓝色房子，再经过黄色房子。蓝色房子不与黄色房子相邻。这些彩色房子的排列有多少种可能？

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：有限制的排列排列分类讨论

难度评级：1140

解答：

蓝色必须在黄色之前且不相邻，所以蓝黄位置只能是 $(1,3)$ 、 $(1,4)$ 、 $(2,4)$ 。

每种情况下，橙色和红色填入剩下两个位置，并且橙色必须在红色之前，因此被唯一确定。三种排列依次为：橙、蓝、红、黄；蓝、橙、红、黄；蓝、橙、黄、红。

共有 $3$ 种排列。

所以正确答案是 B。

Blue must come before yellow but not next to it, so they sit in positions $(1,3)$ or $(1,4)$ or $(2,4)$ .

In each case orange and red fill the two remaining spots with orange before red, which is forced. The three orderings are orange, blue, red, yellow; blue, orange, red, yellow; and blue, orange, yellow, red.

There are $3$ possible orderings.

Thus, B is the correct answer.

2014 AMC 10A 第 4 题

答案：B

难度评级：1140

解答：

其他年份的第 4 题