Problem 30 · 2013 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
Five cars enter a roundabout at the same time, one at each of the five entrances. Each car drives less than a full loop and exactly one car leaves at each of the five exits. In how many different ways can the cars leave the roundabout?
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Answer: B — 44
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Hint 1 of 2
Each car must leave at a different exit and cannot ride a full lap back to its own entry.
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Hint 2 of 2
So no car exits at its own entrance — count arrangements with no fixed point.
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Approach: count derangements of 5
- Label exits by the cars' entry points; a valid assignment sends each car to a different exit, never its own.
- That is a derangement of 5 objects.
- The number of derangements of 5 is 44, so B.
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