Problem 15 · 2015 AMC 8
Medium
Counting & Probability
inclusion-exclusion
At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?
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Answer: D — 99 students.
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Hint 1 of 2
The '29 against both' is the door in: everyone else — 198 − 29 — said yes to at least one issue. That single number is what unlocks the rest.
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Hint 2 of 2
When you add the two 'yes' counts (149 + 119), the people who said yes to both got counted twice. Inclusion-exclusion fixes the overlap: |A ∪ B| = |A| + |B| − |A ∩ B|.
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Approach: inclusion-exclusion, after finding the 'at least one' group
- Against both = 29, so everyone else voted yes to at least one issue: 198 − 29 = 169 = |A ∪ B|.
- Adding 149 + 119 counts the 'both' voters twice, so 149 + 119 − |both| = 169.
- |both| = 149 + 119 − 169 = 99.
- Why this transfers: the moment you add two overlapping groups, you've double-counted their intersection — subtract it once. A quick Venn sketch (two circles in a box of 198) makes the bookkeeping foolproof.
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