There are 12 points in a plane, no three points are collinear except 6 points. How many differen

step 1 Hi, so this is Christian number 45 and the question says there are 12 points in a plane and no t

There are 12 points in a plane, no three points are...

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Key Concepts

Combination Theory

This concept involves counting the number of ways to select a set of items from a larger pool without regard to the order of selection. In problems involving points and triangles, it typically uses the combination formula, often expressed as 'n choose k', where n is the total number of points and k is the number of points taken to form a triangle. This provides the initial count of all possible sets of three points before accounting for any restrictions.

Collinearity in Geometry

Collinearity refers to the condition where three or more points lie on a single straight line. In geometric counting problems, collinear points do not form a valid triangle because they are degenerate, meaning they have no area. Recognizing which sets of points are collinear is essential to accurately exclude these from the total count of triangles.

Exclusion of Degenerate Cases

After calculating the total possibilities using combinations, it is crucial to subtract the cases where the chosen points are collinear. This adjustment ensures that only valid, non-degenerate triangles—those with a defined area—are counted. Understanding and applying this exclusion is key to solving problems where subsets of points might not contribute to the formation of triangles.

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