Problem 17 · 2021 Math Kangaroo
Stretch
Counting & Probability
careful-counting
Five cars participated in a race, starting in the order I, II, III, IV, V. Whenever a car overtook another car, a point was awarded. The cars reached the finish line in the order III, V, I, IV, II. What is the smallest number of points in total that could have been awarded?
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Answer: E — 6
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Hint 1 of 2
The fewest overtakes equals the number of pairs that swap their relative order from start to finish.
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Hint 2 of 2
Compare every pair of cars and count how many ended in the opposite order to how they started.
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Approach: count order-reversed pairs (inversions)
- Each overtake swaps one adjacent pair, so the minimum total equals the number of pairs whose order reversed.
- Start I,II,III,IV,V; finish III,V,I,IV,II.
- Counting all pairs that flipped relative order gives 6 such pairs.
- So the smallest possible total is 6, choice (E).
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