There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is
Text Solution
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To count the number of distinct triangles:
Start with all possible triples of points.
From 12 points, the number of ways to choose any 3 is
Subtract the "invalid" triples that are collinear.
The only collinear sets of three points arise from the single line containing the 5 collinear points.
Number of ways to pick 3 points from those 5 is
Valid triangles = total triples - collinear triples.
Hence, the total number of triangles that can be formed is 
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