2022 AMC 10A Problem 9

Question

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

Answer: D

Concepts:arrangements with restrictions multiplication principle

Difficulty rating: 1220

Small hint:

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

Big hint:

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

Solution:

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.