AMC 8 1997 Problem 9
AMC8 1997 年第 9 题
概率★★☆☆☆
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?
三个名字各不相同的学生排成一列。他们从前到后按字母顺序排列的概率是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
There are 3 ways to pick the first student, 2 ways to pick the second student, and 1 way to pick the last student, for a total of ways to line the students up.
Only 1 of those ways is alphabetical. Thus, the probability is or
Give the students uncreative names like Abby Adams, Bob Bott, and Carol Crock: , , and , replacing the alphabetically first student's name with , and the alphabetically last student's name with . List all the ways they can line up:
, , , , ,
Only 1 of those 6 lists is in alphabetical order, giving an answer of or
We look at the first place in the line; there is only one correct person who belongs in this place, but without the restriction there can be three different people who can go in this spot. So we have a chance of getting the first place right. Now look at the second place; there is only one correct person who belongs there, but without the restriction there can be two different people, as one person is already in the first place. So we have a chance of getting the second place right. Now look at the third place; there is only one correct person, and since all the other spots are filled there is only one person who can be in this place at all. So we have a chance of getting the third place right. Since the total probability depends on the probability of each place having the right person, we multiply these three fractions: . Simplifying, we get that the answer is , or .
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