Problem 9 · 1999 AMC 8
Medium
Counting & Probability
inclusion-exclusion
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Answer: C — 1150 plants.
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Hint 1 of 2
If you just add 500 + 450 + 350, every plant in an overlap gets counted in two beds — so the sum is too big. By how much?
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Hint 2 of 2
A plant shared by two beds is counted twice but should count once, so it's an extra +1. Subtract each overlap once to undo the double-count.
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Approach: inclusion–exclusion: add all, subtract the double-counted overlaps once
- Add the beds: 500 + 450 + 350 = 1300. But a plant in two beds was tallied in both, so each overlap plant is counted one time too many.
- There are 50 + 100 = 150 such doubly-counted plants (and none in all three). Remove the extra copy: 1300 − 150 = 1150 plants.
- This is inclusion–exclusion, and you'll meet it everywhere: |A ∪ B ∪ C| = (sum of parts) − (pairwise overlaps) + (triple overlap). Here the triple overlap is 0, so you only subtract the pairs.
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