Problem 13 · 2007 AMC 8
Medium
Counting & Probability
inclusion-exclusion
Sets A and B, shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.
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Answer: C — 1504.
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Hint 1 of 2
If you add the two circles, the overlap gets counted twice — so |A| + |B| overshoots the union by exactly the intersection. That's the whole equation.
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Hint 2 of 2
Inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|, because the shared part is double-counted once.
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Approach: inclusion-exclusion with equal sets
- Since |A| = |B|, write both as x. The two circles together cover the union plus one extra copy of the overlap: x + x − 1001 = 2007.
- So 2x = 2007 + 1001 = 3008, giving x = 1504.
- Sanity check: each set must be at least as big as the intersection (1001) and at most the union (2007); 1504 sits comfortably between, as it should.
Another way — count the three regions of the diagram:
- The middle (both) holds 1001. The union has 2007, so the two outer-only slivers together hold 2007 − 1001 = 1006.
- Equal sets means the slivers split evenly: 503 each. Then |A| = its sliver + the middle = 503 + 1001 = 1504.
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