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2004 AMC 10A Problem 13

Problem 13 of 25IntermediateCounting & Probability

At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?

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Solution

The number of dancing pairs is 12⋅3=36,12 \cdot 3 = 36, counting from the men’s side. Each woman was in exactly 22 pairs, so the number of women is 362=18. \dfrac{36}{2} = 18. Thus, the correct answer is D.

More practice

Concepts: double counting · counting pairs

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.