Question

在一场射击比赛中，八个泥靶排成三列悬挂：两列各有三个靶，另一列有两个靶。射手必须按以下规则击碎所有靶：

$(1)$ 射手先选择一列，从中击碎一个靶。

$(2)$ 射手必须击碎所选列中剩余的最低靶。

若遵守这些规则，八个靶可以按多少种不同的顺序被击碎？

In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:

$(1)$ The marksman first chooses a column from which a target is to be broken.

$(2)$ The marksman must then break the lowest remaining target in the chosen column.

If the rules are followed, in how many different orders can the eight targets be broken?

答案

答案：560

知识点：多重集排列有限制的排列组合

难度评级：1800

小提示：

每一列中，从下到上的顺序是固定的

Within each column, the bottom-to-top order is forced

大提示：

只用所选列的序列编码击碎顺序，三列的出现次数分别为 $3$ 、 $3$ 和 $2$

Encode an order only by the sequence of chosen columns, with multiplicities $3,$ $3,$ and $2$

解答：

每一枪所选的列一旦确定，该列中要击碎的靶也就确定。因此，每个有效顺序对应于第一列的符号出现三次、第二列的符号出现三次、第三列的符号出现两次的一种排列。这样的排列数为 $\frac{8!}{3!\,3!\,2!}=560\text{。}$

Once the chosen column is known at each shot, the target within that column is forced. Thus every valid order corresponds to an arrangement of three symbols from the first column, three from the second, and two from the third. The number of such arrangements is $\frac{8!}{3!\,3!\,2!}=560.$

1990 AIME 第 8 题

答案：560

难度评级：1800

小提示：

大提示：

解答：

其他年份的第 8 题