Question

如图，一个长方形被分成 $5$ 个区域。每个区域要涂成纯色，可选颜色为红、橙、黄、蓝、绿。相邻接的区域颜色必须不同，颜色可以重复使用。共有多少种不同涂色方法？

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：有限制的排列乘法原理

难度评级：1220

解答：

左下区域有 $5$ 种颜色可选。左上区域与它相邻，因此有 $4$ 种选择。中间下方区域同时与前两个区域相邻，因此有 $3$ 种选择。

右上区域也与前面两个区域相邻，因此有 $3$ 种选择。最后，右下区域同样有 $3$ 种选择。

把这些选择相乘，得到 $5\cdot4\cdot3\cdot3\cdot3=540$ 种涂色方法。

所以正确答案是 D。

There are $5$ choices for the color of the bottom left rectangle. This forces there to be $4$ choices for the top left rectangle. The middle bottom rectangle touches both of the previous ones, so there are $3$ color options for this rectangle.

The rectangle in the top right is also limited to $3$ colors since it touches the two previous rectangles. Finally, the rectangle in the bottom right also has $3$ color options.

Multiplying these together, we get $5\cdot4\cdot3\cdot3\cdot3=540$ total colorings.

Thus, D is the correct answer.

2022 AMC 10A 第 9 题

答案：D

难度评级：1220

解答：

其他年份的第 9 题