Question

Flatville 的自行车牌照每个含有三个字母。第一个字母从集合 $\{C,H,L,P,R\}$ 中选，第二个从 $\{A,I,O\}$ 中选，第三个从 $\{D,M,N,T\}$ 中选。

当 Flatville 需要更多牌照时，他们增加了两个新字母。这两个新字母可以都加入同一个集合，也可以分别加入两个不同集合。加入两个字母后，最多能多制作多少个额外牌照？

Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\},$ the second from $\{A,I,O\},$ and the third from $\{D,M,N,T\}.$

When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of additional license plates that can be made by adding two letters?

$24$

$30$

$36$

$40$

$60$

答案：D

知识点：乘法原理最优化分类讨论

难度评级：1330

解答：

目前可以制作的牌照数为 $5 \cdot 3 \cdot 4 = 60$

如果两个字母都加入第一个集合，则有个可能牌照。 $7 \cdot 3 \cdot 4 = 84$

如果两个字母都加入第二个集合，则有个牌照。 $5 \cdot 5 \cdot 4 = 100$

如果两个字母都加入第三个集合，则有个选择。 $5 \cdot 3 \cdot 6 = 90$

如果一个字母加入第一个集合，另一个加入第二个集合，则有个牌照。 $6 \cdot 4 \cdot 4 = 96$

如果一个加入第一个集合，另一个加入第三个集合，则有个可能牌照。 $6 \cdot 3 \cdot 5 = 90$

最后，如果字母加入第二和第三个集合，则有个牌照。 $5 \cdot 4 \cdot 5 = 100$

最大可以达到 $100$ 个牌照，因此额外增加 $100 - 60 = 40$ 个。

所以正确答案是 D。

There are currently $5 \cdot 3 \cdot 4 = 60$ license plates that can be made.

If both letters are added to the first set, then there are $7 \cdot 3 \cdot 4 = 84$ possible plates.

If they are both added to the second, there are $5 \cdot 5 \cdot 4 = 100$ plates.

If they are added to the third, there are $5 \cdot 3 \cdot 6 = 90$ choices.

If one is added to the first set and the other to the second set, there are $6 \cdot 4 \cdot 4 = 96$ plates.

If the other is added to the third set, we get $6 \cdot 3 \cdot 5 = 90$ possible plates.

Finally, if the letters are added to the second and third sets, there are $5 \cdot 4 \cdot 5 = 100$ plates.

We see that $100$ is the greatest number of plates that we can achieve. This is an additional $100 - 60 = 40$ plates.

Thus, D is the correct answer.

1999 AMC 8 第 15 题

答案：D

难度评级：1330

解答：

其他年份的第 15 题