Question

一位农民的长方形田地被分成如下图所示的 $2$ 乘 $2$ 网格，共 $4$ 个长方形区域。每个区域将种一种作物：玉米、小麦、大豆或土豆。农民不想让玉米和小麦种在任何共边的两个区域，也不想让大豆和土豆种在任何共边的两个区域。在这些限制下，农民有多少种方法为四个区域选择作物？

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?

$12$

$64$

$84$

$90$

$144$

答案：C

知识点：有限制的排列分类讨论

难度评级：1540

解答：

分为 $2$ 种情况讨论右上和左下这两个对角区域。

情况 $1$ ：右上和左下区域种同一种作物。

这两个区域所种的作物有 $4$ 种选择，另外两个区域各有 $3$ 种选择，所以共有种。 $4 \cdot 3 \cdot 3 = 36$

情况 $2$ ：右上和左下区域种不同作物。

先选左上区域有 $4$ 种，右上有 $3$ 种，左下有 $2$ 种，最后右下区域有种可行选择。因此共有种。 $4 \cdot 3 \cdot 2 \cdot 2 = 48$

两种情况合计 $36 + 48 = 84$ 种。

所以正确答案是 C。

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 第 18 题

答案：C

难度评级：1540

解答：

其他年份的第 18 题