Question

A farmer's rectangular field is partitioned into $2$ 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?

$12$

$64$

$84$

$90$

$144$

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 which crop is in the top-left section. Then there are $3$ options for the top-right and $2$ for the bottom-left. This leaves $2$ options for the bottom-right section. 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

Solution:

Problem 18 in Other Years