AMC 8 2015 Problem 11
AMC8 2015 年第 11 题
概率★★★☆☆
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
在 Mathland 这个小国家，所有汽车的牌照都有 4 个符号。第一个符号必须是个韵母（A,E,I,O 或 U），第二个和第三个符号必须是 21 个非韵母中不同的两个字母，第四个必须是一个数字
（0 到 9），如果符号都是按照这些规则随机选择的，那么牌照上显示的是“AMC8”概率是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
There is one favorable case, which is the license plate says "AMC8." We must now find how many total cases there are. There are 5 choices for the first letter (since it must be a vowel), 21 choices for the second letter (since it must be of 21 consonants), 20 choices for the third letter (since it must differ from the second letter), and 10 choices for the digit. This leads to total possible license plates. Therefore, the probability of a license plate saying "AMC8" is .
The probability of choosing A as the first letter is . The probability of choosing next is . The probability of choosing C as the third letter is (since there are 20 other consonants to choose from other than M). The probability of having 8 as the last number is . We multiply all these to obtain .
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