2004 AMC 12A Problems/Problem 13

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Problem

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 $S$ ?

$\text {(A)}\ 8 \qquad \text {(B)}\ 20 \qquad \text {(C)}\ 24 \qquad \text {(D)}\ 27\qquad \text {(E)}\ 36$

Solution

Solution 1

Let's count them by cases:

Case 1: The line is horizontal or vertical, clearly $3 \cdot 2 = 6$ .
Case 2: The line has slope $\pm 1$ , with $2$ through $(0,0)$ and $4$ additional ones one unit above or below those. These total $6$ .
Case 3: The only remaining lines pass through two points, a vertex and a non-vertex point on the opposite side. Thus we have each vertex pairing up with two points on the two opposites sides, giving $4 \cdot 2 = 8$ lines.

These add up to $6+6+8=20\ \mathrm{(B)}$ .

Solution 2

There are ${9 \choose 2} = 36$ ways to pick two points, but we've clearly overcounted all of the lines which pass through three points. In fact, each line which passes through three points will have been counted ${3 \choose 2} = 3$ times, so we have to subtract $2$ for each of these lines. Quick counting yields $3$ horizontal, $3$ vertical, and $2$ diagonal lines, so the answer is $36 - 2(3+3+2) = 20$ distinct lines.

2004 AMC 12A (Problems • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

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