Number Of Triangles From Points

A triangle is determined by choosing 3 non-collinear points, so when no three points are collinear the count is simply a combination, a clean JEE Advanced geometry-counting result. With 10 points and no three collinear, every selection of 3 points forms a valid triangle, and the number of such selections is C(10, 3) = (10 × 9 × 8)/(3 × 2 × 1) = 120. The non-collinearity condition ensures no chosen triple is degenerate. Option 720 = P(10,3) counts ordered triples, overcounting each triangle 6 times. Option 30 miscomputes the combination. Option 100 has no combinatorial basis. Hence there are 120 triangles. Plausibility check: dividing the ordered triple count 10 × 9 × 8 = 720 by 3! = 6 to remove vertex orderings gives 120, matching C(10,3) and confirming that each unordered triple yields exactly one triangle. Distributing distinct objects into distinct boxes assigns each object an independent choice, so the count is an exponential and coincides exactly with the number of functions between the corresponding sets. Recognizing this equivalence between distributions and functions unifies counting problems with the function-counting results from sets and relations, a connection examiners value across chapter boundaries.