Problem 22 · 2017 Math Kangaroo
Stretch
Counting & Probability
careful-counting
There are 10 kangaroos in a row, as seen in the picture. Two kangaroos that are standing next to each other and can see each other are allowed to change places by hopping past each other. This is carried out until no more jumps are allowed. How often do two kangaroos swap places?
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Answer: C — 18
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Hint 1 of 2
A swap happens for each pair of kangaroos that start facing each other but in the wrong order.
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Hint 2 of 2
Count the facing pairs that must pass each other — that is the number of swaps.
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Approach: count the pairs that must cross
- A swap happens exactly once for each pair where a right-facing kangaroo starts somewhere to the left of a left-facing kangaroo, since those two must pass each other.
- So count, for every right-facing kangaroo, how many left-facing kangaroos stand to its right, and add these up.
- For the arrangement shown that total is 18 swaps.
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