2004 AMC 12A Problem 13

Question

Let

S

be the set of points

(a, b)

in the coordinate plane, where each of

a

and

b

may be

-1,

0,

or

1.

How many distinct lines pass through at least two members of

S?

8

20

24

27

36

Start from the

\binom{9}{2}

pairs of points, each of which determines a line

Some lines pass through three collinear points and get counted

\binom{3}{2} = 3

times

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.

Problem 13 in other years