Problem 9 · 1997 AJHSME
Medium
Counting & Probability
counting-arrangements
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front to back?
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Answer: C — 1/6.
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Hint 1 of 2
Every possible lineup is equally likely, and only ONE of them is alphabetical. So the question is really 'how many lineups are there in total?'
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Hint 2 of 2
Probability of one specific arrangement = 1 ÷ (total number of arrangements). Count the orderings of n distinct items as n × (n−1) × … × 1.
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Approach: favorable orders over all equally likely orders
- Three different people can line up in 3 × 2 × 1 = 6 distinct orders (3 choices for front, 2 for middle, 1 left for back).
- Exactly one of those orders is alphabetical, and all 6 are equally likely, so the probability is 1/6.
Another way — pure symmetry — no counting:
- Whatever order they happen to stand in, relabeling can't favor one arrangement over another, so each of the possible orders is equally likely.
- Alphabetical is just one of those orders. With 3 names there are 3! = 6 orders, so its share is 1/6 — no listing required.
Mark:
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