Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #21
Grade 7 counting
Pick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting valid triangles directly would mean checking every triple for collinearity — a lot of cases. The complement is far smaller: of all triples of 3 dots, only the collinear ones are NOT triangles. So Tool #16 (Complement) fits: count every triple, then subtract the few bad ones. Tool #1 (Draw a Diagram) is the natural sidekick — sketching the 2 × 3 grid makes the only straight lines through 3 dots jump out (the two horizontal rows). Tool #2 (Systematic List) confirms there are no other 3-dot lines by walking through the diagonals.
Count all triples of dots
Count every way to pick 3 of the 6 dots with order ignored — the combination C(6,3) gives 20 possible trios.
This is the universe in the complement plan: every possible triple of dots, valid triangle or not.
7.SP.C.8Change Focus Count The ComplementLook for straight lines
Sketch the 2 × 3 grid and hunt for any 3 dots on a single straight line — the only candidates are the rows, the columns, and the diagonals.
A picture of the grid makes the straight-line triples visible at a glance — the rows are the obvious culprits.
5.G.A.1Draw A DiagramList the collinear triples
Each column holds only 2 dots and every diagonal hits at most 2, so the only 3-dot straight lines are the two horizontal rows.
Going row → column → diagonal in order guarantees no collinear triple is missed and none is double-counted.
7.SP.C.8Make A Systematic ListSubtract the straight lines
Subtract the 2 collinear rows from the 20 trios, leaving 18 genuine triangles.
Complement counting: valid = total - bad. The 2 bad triples shave the 20 choices down to the 18 real triangles.
The number of triangles equals the number of ways to choose three of the six dots, minus the number of those choices whose three dots lie on one straight line.
▸ Why?
The chosen trios of dots split into just two groups with nothing shared and nothing left over — those that form a triangle and those whose three dots lie on one straight line — so taking the straight-line group out of all the trios leaves exactly the triangles.
▸ Why?
A trio falls in the straight-line group exactly when its three dots are collinear: then the middle dot opens into a straight 180° angle instead of a bent corner, so no triangle forms and the two groups never overlap.
▸ Why?
Once the trios are cut into these two non-overlapping groups that together make up every trio, the count of one group is the whole count minus the count of the other.
When most picks of 3 are valid and only a few are not, count everything and subtract the bad ones — here C(6, 3) = 20 total minus the 2 flat rows leaves 18 real triangles.
- Count all triples of dots
- Look for straight lines
- List the collinear triples
- Subtract the straight lines
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