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 ?
- (a) 16
- (b) 18
- (c) 20 ✓ UPSC's answer
- (d) 24
Why the answer is (c)
• There are 6 points in all: one on AB, one on CD, two on BC and two on DA.
• Any 3 of the 6 can be chosen in 6C3 = 20 ways.
• Three points fail to make a triangle only if they lie on one straight line; points on different sides of a square are never collinear, and no side holds three points.
• So all 20 choices give triangles, option (c).
Why the other options are wrong
- (a) 16
- No three of the points are collinear, so none of the 20 selections is lost.
- (b) 18
- All 20 selections form triangles, not 18.
- (d) 24
- Only 20 selections of 3 points exist, so 24 is impossible.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.