Question

一个长方形如图被分成 $5$ 个区域。每个区域要涂成一种纯色，可选颜色为红、橙、黄、蓝、绿。相接触的区域必须涂成不同颜色，颜色可以重复使用。共有多少种不同的涂色方法？

A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?

$120$

$270$

$360$

$540$

$720$

答案：D

知识点：图论乘法原理

难度评级：1380

解答：

底部中间的区域与其他四个区域都共边。先给它涂色，有 $5$ 种方法。

左上区域与它相邻，有 $4$ 种选择。其余三个区域各与两个已经涂好的区域相邻，而这两个区域颜色不同，所以各有 $3$ 种选择。

总数为 $5\cdot4\cdot3\cdot3\cdot3=540$ 。

因此，正确答案是 D。

The bottom-middle region shares a border with all four other regions. Color it first in $5$ ways.

The top-left region borders it, giving $4$ choices. Each of the three remaining regions borders exactly two already-colored regions, which have different colors, leaving $3$ choices apiece.

The total is $5\cdot4\cdot3\cdot3\cdot3=540.$

Thus, the correct answer is D.

2022 AMC 12A 第 7 题

答案：D

难度评级：1380

解答：

其他年份的第 7 题