Problem 26 · 2014 Math Kangaroo
Stretch
Counting & Probability
complementary-countingcareful-counting
Every group of three vertices of a cube forms a triangle. How many such triangles are there whose vertices do not all lie on the same face of the cube?
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Answer: C — 32
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Hint 1 of 2
Count all triangles from the 8 vertices, then remove the 'flat' ones.
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Hint 2 of 2
A triangle is bad exactly when all three vertices lie on one face.
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Approach: total triangles minus the same-face ones
- All triangles: C(8,3) = 56.
- Each of the 6 faces has C(4,3) = 4 same-face triangles, so 6 × 4 = 24 are bad.
- Good triangles: 56 − 24 = 32.
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