Problem 23 · 2015 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
In how many ways can the three kangaroos be placed in three different squares of the row of 7 squares so that no kangaroo has an immediate neighbour?
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Answer: D — 10
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Hint 1 of 3
The kangaroos look the same, so you are really just choosing which 3 of the 7 squares are filled, with no two filled squares touching.
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Hint 2 of 3
Put the 4 empty squares in a row and notice the 5 gaps around them (one before, three between, one after).
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Hint 3 of 3
Dropping a kangaroo into a gap automatically keeps them apart, so just count ways to pick 3 of those 5 gaps.
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Approach: place the empty squares first, then drop kangaroos into the gaps so they can't touch
- Since the kangaroos are identical, we just choose 3 of the 7 squares to fill, with no two chosen squares next to each other.
- Line up the 4 empty squares: _ ▢ _ ▢ _ ▢ _ ▢ _. The 5 underscores are the safe spots, and any kangaroo placed in a gap is automatically separated from the others.
- Choosing 3 of these 5 gaps, leaving 2 empty, the gap-pairs left out are 12,13,14,15,23,24,25,34,35,45 — that's 10 ways.
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