2004 AMC 10A Problem 5

P1-10~2 minPrint

Basic Probability Basic Counting Organized listing / direct enumeration

A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

Show solution without answering

Key insight

There are 84 ways to pick 3 of the 9 points, and only 8 of them lie on a line: the 3 rows, the 3 columns and the 2 diagonals.

Solution

Probability is (number of good choices) divided by (number of all choices), so count both.

1. All choices. The grid has $9$ points. The number of ways to choose $3$ of them, when order does not matter, is
$\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84.$
2. Good choices. Three points of this grid lie on one straight line only in these cases:
- a full row: $3$ ways,
- a full column: $3$ ways,
- a full diagonal corner to corner: $2$ ways.

Any other line (for example a steeper slant) passes through at most $2$ of the grid points. So there are $3 + 3 + 2 = 8$ good choices.
3. Probability.
$\frac{8}{84} = \frac{2}{21}.$

The answer is $\boxed{\textbf{(C)}\ \frac{2}{21}}$ .

Why this works

When every choice is equally likely, count the total with a combination and then list the good cases by type. On a $3 \times 3$ grid, the lines of three are exactly the eight ways to win at tic-tac-toe, which is an easy picture to remember.

Concise solution

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

The grid has $9$ points, so the number of equally likely three-point sets is
$\binom{9}{3} = \frac{9 \cdot 8 \cdot 7}{6} = 84.$

A line through three of these points must be a full row, a full column, or a main diagonal, since any other line hits at most two grid points. That gives $3 + 3 + 2 = 8$ collinear triples.

The probability is $\dfrac{8}{84} = \dfrac{2}{21}$ , so the answer is $\boxed{\textbf{(C)}\ \frac{2}{21}}$ .

Review in the AoPS books

Introduction to Counting & Probability, Chapter 7: Introduction to Probability
Introduction to Counting & Probability, Chapter 1: Counting Is Arithmetic
Prealgebra, Chapter 15: Problem-Solving Strategies (§15.2)
Introduction to Counting & Probability, Chapter 8: Basic Probability Techniques