Problem 25 · 2019 AMC 8
Hard
Counting & Probability
stars-and-barscareful-counting
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
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Answer: C — 190 ways.
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Hint 1 of 2
The "at least 2 each" rule is the only obstacle — so satisfy it up front. Hand everyone their 2 apples first; whatever's left can be split with no rules at all.
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Hint 2 of 2
Now it's "split N identical apples among 3 people, zero allowed." That's stars and bars: lay out the apples in a row and choose where 2 dividers go.
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Approach: give the minimum first, then unrestricted stars and bars
- Pre-give 2 apples to each person (6 used), so the floor is automatically met. That leaves 18 apples to hand out among the three with no lower limit — the hard constraint is gone.
- Picture the 18 apples in a row; placing 2 dividers among them splits them into the three shares (a person can get 0). Choosing 2 divider slots out of 18 + 2 = 20 positions gives C(20, 2) = 190.
- Why this transfers: a "each gets at least k" condition is removed by pre-allotting k to everyone, converting it to the standard zero-allowed stars-and-bars count C(n + r − 1, r − 1).
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