Problem 30 · 2013 Math Kangaroo
Stretch
Counting & Probability
Logic & Word Problems
careful-countingcasework
Four cars drive into a roundabout at the same moment, each from a different direction (see diagram). No car drives all the way around the roundabout, and no two cars leave by the same exit. In how many different ways can the cars exit the roundabout?
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Answer: A — 9
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Hint 1 of 2
No car exits where it entered (that would be a full loop), and all four exits are different.
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Hint 2 of 2
That is exactly a permutation of four things with no item in its own place.
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Approach: count derangements of 4
- Each car must leave by a different exit, and not its own entrance — a permutation with no fixed point.
- The number of derangements of 4 items is 9.
- So the cars can exit in 9 different ways.
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