Question

Let \( S \) be the set of all points with coordinates \( (x, y, z) \), where \( x, y, z \) are each chosen from the set \([0, 1, 2]\). How many equilateral triangles have all their vertices in \( S \)?

• A. 72
• B. 76
• C. 80
• D. 84

Hint:

When counting the number of equilateral triangles in a discrete set of points, use combinatorics and geometric properties to identify valid triangles.

Answer 1

The Correct Option is A

We are given the coordinates \((x, y, z)\) where \( x, y, z \in \{0, 1, 2\} \). The total number of points in the set \( S \) is \( 3 \times 3 \times 3 = 27 \). An equilateral triangle is formed by selecting any 3 points that are equidistant from each other. The number of such triangles that can be formed from the set of 27 points is 72.