ABCD is a square. One point on each of AB and CD; and two distinct points on each of BC and DA are chosen. How many distinct triangles can be drawn using any three points as vertices out of these six points?
Answer & explanation
Answer: (c) 20
Three points fail to make a triangle only if they lie on one straight line. No side of the square carries more than two of the six points, so no three are in a line and every choice of three points gives a triangle: C(6, 3) = 20.
- Points: 1 on AB, 1 on CD, 2 on BC and 2 on DA — six in all.
- A set of three points forms no triangle only when all three lie on one side; no side has more than two points.
- Number of triangles = C(6, 3) = (6 × 5 × 4)/(3 × 2 × 1) = 20.
Remember · Triangles from points = C(n, 3) minus collinear triples; check each line for three or more points.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·