2021 AMC 10A Fall Problem 18

Question

A farmer’s rectangular field is partitioned into a

2

by

2

grid of

4

rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?

12

64

84

90

144

Answer: C

Concepts:arrangements with restrictions casework

Difficulty rating: 1540

Small hint:

Case on whether the two diagonal sections are the same crop

Big hint:

Each shared-border restriction removes one crop choice

Solution:

There are

2

cases.

Case

1:

the top-right and bottom-left sections have the same crop

There are

4

options for which crop is in those sections. The other two sections have

3

options for the crop since each crop has one restriction for crops next to it. This gives us

4 \cdot 3 \cdot 3 = 36

combinations.

Case

2:

the top-right and bottom-left sections have different crops

There are

4

options for the top-right crop and

3

different options for the bottom-left crop. Each of the other two sections borders both of these crops, leaving

2

choices for each. This gives us another

4 \cdot 3 \cdot 2 \cdot 2 = 48

configurations.

In total, there are

36 + 48 = 84

combinations.

Thus, C is the correct answer.

2021 AMC 10A Fall Problem 18

Answer: C

Difficulty rating: 1540

Small hint:

Big hint:

Solution:

Problem 18 in other years