Question

有多少种方法把 $3$ 枚不可区分的红色筹码、 $3$ 枚不可区分的蓝色筹码和 $3$ 枚不可区分的绿色筹码放入一个 $3 \times 3$ 方格的格子中，使得任意两个同色筹码都不在上下或左右方向相邻？

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?

$12$

$18$

$24$

$30$

$36$

答案：E

知识点：有限制的排列分类讨论

难度评级：1820

解答：

先把三种颜色命名为 $\binom42=6$ 和。这 $3$ 种颜色与红、蓝、绿对应有种方式，所以最后要乘以。

设中心格为。另外两个不能与中心相邻，只能放在角格，并且要么在同一条对角线上，要么在同一边的两个角上。第一种情形有 $2$ 种，因为有条对角线；第二种情形有种，因为有条边。 $3\binom42\cdot2=36.$

在每种情形中，剩余格子中和的位置被唯一确定，否则会产生同色相邻。因此固定和时共有种安排，乘上颜色名称的对应方式，总数为。所以正确答案是 E。

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.

2021 AMC 10A Spring 第 25 题

答案：E

难度评级：1820

解答：

其他年份的第 25 题