2004 AMC 10A Problem 13

P11-15~2 minPrint

Basic Counting Ratios, Percents & Averages Map the objects to something easier to count

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?

Show solution without answering

Key insight

Count dance pairs two ways. The men make 12 times 3 = 36 pairs; each woman is in 2 pairs, so there are 18 women.

Solution

Think about the list of every (man, woman) pair who danced together. Count that list two different ways.

1. Counting by the men. There are $12$ men, and each is in $3$ pairs:
$12 \times 3 = 36 \text{ pairs}.$
2. Counting by the women. Each woman is in $2$ pairs. If there are $w$ women, the list has $2w$ pairs.
3. It is the same list, so the two counts agree:
$2w = 36, \qquad w = 18.$

The answer is $\boxed{\textbf{(D)}\ 18}$ .

Why this works

This is called double counting: one collection (the dance pairs) is counted once from each side, and both counts must give the same number. The same trick works for handshakes, games in a tournament, and people on committees.

Alternative approach

Use a small group. $2$ men make $2 \times 3 = 6$ pairs, and $3$ women also make $3 \times 2 = 6$ pairs. So men and women come in the ratio $2 : 3$ . With $12$ men, that is $18$ women.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Consider the set of all (man, woman) pairs who danced together, and count it from each side.

From the men's side: $12$ men, each in $3$ pairs, so there are $12 \cdot 3 = 36$ pairs.

From the women's side: if there are $w$ women, each in $2$ pairs, the same set has $2w$ pairs.

Equating, $2w = 36$ , so $w = 18$ .

The answer is $\boxed{\textbf{(D)}\ 18}$ .

Review in the AoPS books

Introduction to Counting & Probability, Chapter 1: Counting Is Arithmetic
Introduction to Algebra, Chapter 6: Ratios and Percents
Intermediate Counting & Probability, Chapter 4: Constructive Counting and 1-1 Correspondences (§4.4, §4.5, §4.6)
Introduction to Counting & Probability, Chapter 2: Basic Counting Techniques