Problem 21 · 2005 AMC 8
Medium
Counting & Probability
collinear-exclusion
How many distinct triangles can be drawn using three of the dots below as vertices?
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Answer: C — 18.
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Hint 1 of 2
Count generously first: every choice of 3 dots gives a triangle — unless the 3 happen to lie on one straight line. So count all triples, then throw out the flat ones.
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Hint 2 of 2
The only way 3 of these 6 dots line up is if all three sit in the same row. How many all-in-one-row triples are there?
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Approach: count all triples, subtract the collinear ones
- Choosing 3 dots from 6 in any way: C(6,3) = 20 triples.
- Three dots fail to make a triangle only when they're collinear. Here the dots form two rows of 3, so the lone collinear triples are the full top row and the full bottom row — just 2 of them.
- Triangles: 20 − 2 = 18.
- Why this 'count all, subtract bad' works: it's far easier to count every selection and remove the few degenerate ones than to count valid triangles directly. The trap answer 20 (choice D) is forgetting that collinear triples aren't triangles.
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