AMC12 2014 A

AMC12 2014 A · Q3

AMC12 2014 A · Q3. It mainly tests Casework, Logic puzzles.

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?

走在简街上，拉尔夫经过了四栋连续的房子，每栋房子漆成不同的颜色。他经过橙色房子在红色房子之前，经过蓝色房子在黄色房子之前。蓝色房子不紧邻黄色房子。彩色房子的可能排列有多少种？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 6 6

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let's use casework on the yellow house. The yellow house $(\text{Y})$ is either the $3^\text{rd}$ house or the last house. Case 1: $\text{Y}$ is the $3^\text{rd}$ house. The only possible arrangement is $\text{B}-\text{O}-\text{Y}-\text{R}$ Case 2: $\text{Y}$ is the last house. There are two possible arrangements: $\text{B}-\text{O}-\text{R}-\text{Y}$ $\text{O}-\text{B}-\text{R}-\text{Y}$ The answer is $1+2=\boxed{\textbf{(B) } 3}$

按黄色房子位置分类讨论。黄色房子$(\text{Y})$要么是第$3^\text{rd}$栋，要么是最后一栋。情况1：$\text{Y}$是第$3$栋房子。唯一可能的排列是 $\text{B}-\text{O}-\text{Y}-\text{R}$ 情况2：$\text{Y}$是最后一栋房子。有两种可能的排列： $\text{B}-\text{O}-\text{R}-\text{Y}$ $\text{O}-\text{B}-\text{R}-\text{Y}$ 答案是$1+2=\boxed{\textbf{(B) } 3}$

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