2021 AMC 10A Spring Problem 25

Question

How many ways are there to place

3

indistinguishable red chips,

3

indistinguishable blue chips, and

3

indistinguishable green chips in the squares of a

3 \times 3

grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?

12

18

24

30

36

Answer: E

Concepts:arrangements with restrictions casework

Difficulty rating: 1820

Small hint:

First choose the color in the center and the two corners occupied by that color

Big hint:

Once those three chips are placed, the other two colors are forced up to interchange

Solution:

Choose the center color in

3

ways. Its other two chips cannot occupy any edge-middle square, because those squares are adjacent to the center. Thus they must occupy two of the four corners, which can be chosen in

\binom42=6

ways.

For either possible corner pattern—two opposite corners or two corners on the same side—the adjacency conditions force the remaining two colors up to interchanging them. Hence there are

2

completions for each choice of the center color and its two corners. The total is

3\binom42\cdot2=36.

Thus, E is the correct answer.