AMC10 2022 A

AMC10 2022 A · Q9

AMC10 2022 A · Q9. It mainly tests Basic counting (rules of product/sum), Casework.

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?

一个矩形被分成 $5$ 个区域，如图所示。每个区域要涂成纯色——红、橙、黄、蓝或绿——使得相邻区域颜色不同，且颜色可以重复使用。有多少种不同的涂色方案？

(A) 120 120

(B) 270 270

(D) 540 540

(E) 720 720

Answer

Correct choice: (D)

正确答案：（D）

Solution

The top left rectangle can be $5$ possible colors. Then the bottom left region can only be $4$ possible colors, and the bottom middle can only be $3$ colors since it is next to the top left and bottom left. Similarly, we have $3$ choices for the top right and $3$ choices for the bottom right, which gives us a total of $5\cdot4\cdot3\cdot3\cdot3=\boxed{\textbf{(D) }540}$.

左上矩形有 $5$ 种颜色选择。然后左下区域只有 $4$ 种颜色选择，底中区域由于紧邻左上和左下只有 $3$ 种颜色选择。类似地，右上和右下各有 $3$ 种选择，总共 $5\cdot4\cdot3\cdot3\cdot3=\boxed{\textbf{(D) }540}$。

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