AMC 8 1999 Problem 15
AMC8 1999 年第 15 题
基本计数原理★★★☆☆
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?
Flatville 的自行车牌照各包含三个字母。第一个从 {C,H,L,P,R} 中选择，第二个从 {A,I,O} 中选择，第三个从 {D,M,N,T} 中选择。当 Flatville 需要更多牌照时，他们新增了两个新字母。这两个新字母可以都加入一个集合，或者一个字母加入一个集合、另一个加入另一个集合。通过添加两个字母，最多可以增加多少额外的牌照？
- A.24
- B.30
- C.36
- D.40
- E.60
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
There are currently 5 choices for the first letter, 3 choices for the second letter, and 4 choices for the third letter, for a total of license plates.
Adding 2 letters to the start gives plates.
Adding 2 letters to the middle gives plates.
Adding 2 letters to the end gives plates.
Adding a letter to the start and middle gives plates.
Adding a letter to the start and end gives plates.
Adding a letter to the middle and end gives plates.
You can get at most 100 license plates total, giving an additional 100 - 60 = 40 plates, making the answer
Using the same logic as above, the number of combinations of plates is simply the product of the size of each set of letters.
In general, when three numbers have the same fixed sum, their product will be maximal when they are as close together as possible. This is a 3D analogue of the fact that a rectangle with fixed perimeter maximizes its area when the sides are equal (ie when it becomes a square). In this case, no matter where you add the letters, there will be 5 + 3 + 4 + 2 = 14 letters in total. If you divide them as evenly as possible among the three groups, you get 5, 5, 4, which is a possible situation.
As before, the answer is , and the correct choice is
Before new letters were added, five different letters could have been chosen for the first position, three for the second, and four for the third. This means that plates could have been made.
If two letters are added to the second set, then plates can be made. If one letter is added to each of the second and third sets, then plates can be made. None of the other four ways to place the two letters will create as many plates. So, 100-60=40 ADDITIONAL plates can be made.So the correct choice is
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