2004 AMC 12A Problem 13

Problem 13 of 25IntermediateGeometry Counting & Probability

Let

S

be the set of points

(a, b)

in the coordinate plane, where each of

a

and

b

may be

-1,

0,

1.

How many distinct lines pass through at least two members of

S?

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Solution

There are

\binom{9}{2} = 36

pairs of points, and each pair determines a line. However, there are three horizontal, three vertical, and two diagonal lines that each pass through three collinear points of

S.

Each such line is counted

3

times, an overcount of

2

per line. With

8

such lines, the number of distinct lines is

36 - 2 \cdot 8 = 20.

Thus, the correct answer is B.

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