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 chosen at random are subject to these conditions, what is the probability that the plate will read "$AMC8$"?

A. $\frac{1}{22050}$

B. $\frac{1}{21000}$

C. $\frac{1}{10500}$

D. $\frac{1}{2100}$

E. $\frac{1}{1050}$

1. 2015 AMC 8 官方解析

There is one favorable case, which is the license plate that 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 number. This leads to $5 \cdot 21 \cdot 20 \cdot 10 = 21000$ total possible license plates. That means the probability of a license plate saying "$AMC8$" is $\frac{1}{21000}$.