2022 AMC 10A 第 9 题

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

小提示：

按照每个新区域的相邻区域都已确定的顺序给区域涂色

Color the regions in an order where each new region has known neighbors

大提示：

确定前两个区域的颜色后，剩下的每个区域都有三种选择

After the first two colors, each remaining region has three choices

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

小提示：

大提示：

解答：

其他年份的第 9 题