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Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Two ordered pairs $(a, b)$ and $(c, d)$ , where $a, b, c, d$ are real numbers, form a basis of the coordinate plane if $a d \neq b c$ . Determine the number of ordered quadruples $(a, b, c, d)$ of integers between $1$ and $3$ inclusive for which $(a, b)$ and $(c, d)$ form a basis for the coordinate plane.

Solution

Solution:

Any pair of distinct points will form a basis except when $(a, b)$ and $(c, d)$ are both from $\{(1,1),(2,2),(3,3)\}$ , so the answer is $9 \cdot 8 - 3 \cdot 2 = 66$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.