2004 AMC 10A Problem 13

Question

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?

8

12

16

18

24

Answer: D

Concepts:double counting counting pairs

Difficulty rating: 1190

Small hint:

Count the man-woman dancing pairs in two ways

Big hint:

There are

12 \cdot 3

pairs, and each woman accounts for

2

of them

Solution:

The number of dancing pairs is

12 \cdot 3 = 36,

counting from the men’s side. Each woman was in exactly

2

pairs, so the number of women is

\dfrac{36}{2} = 18.

Thus, the correct answer is D.