2021 AMC 10A Fall 第 18 题

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 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

答案：C

知识点：有限制的排列分类讨论

难度评级：1540

小提示：

按两块对角区域是否种同一种作物分类。

Case on whether the two diagonal sections are the same crop

大提示：

每个共边限制都会排除一种作物选择。

Each shared-border restriction removes one crop choice

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。因为每种作物都恰有一种不能与它相邻的作物，所以另外两个区域各有

3

种选择。因此共有

4 \cdot 3 \cdot 3 = 36

种。

情况

2

：右上和左下区域种不同作物。

右上区域有

4

种选择，左下区域有

3

种与之不同的选择。另外两个区域都与这两块区域共边，所以各剩

2

种选择。因此这一类共有

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

种。

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

小提示：

大提示：

解答：

其他年份的第 18 题