UPSC Prelims 2006 Q101: Science & Technology – Quantitative Aptitude
There are three parallel straight lines. Two points A and B are marked on the first line,
points C and D are marked on the second line, and points E and F are marked on the third line.
Each of these six points can move to any position on its respective straight line.
Consider the following statements:
1. The minimum number of triangles that can be formed by joining these points is zero.
2. The maximum number of triangles that can be formed by joining these points is twenty.
Which of the statements given above is/are correct?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Answer: (c) Both 1 and 2
Explanation
Triangles from six points on three parallel lines
Two points lie on each of three parallel lines: A, B; C, D; E, F.
| Statement | Verdict | Reason |
|---|---|---|
| 1. The minimum number of triangles is zero | True | If A and B coincide, C and D coincide, E and F coincide, and these three points lie on one transversal, no triangle forms |
| 2. The maximum number of triangles is twenty | True | There are C(6, 3) = 20 triples, and every triple can form a triangle when points are distinct and not collinear |
Both statements are correct.
Answer: (c) Both 1 and 2.
Why each option is right or wrong
- (a) Incorrect. Statement 2 is also true, since up to 20 triangles are possible.
- (b) Incorrect. Statement 1 is also true, since the points can be placed to give no triangle.
- (c) CORRECT. The minimum is zero (points coincide or are collinear) and the maximum is C(6,3) = 20.
- (d) Incorrect. Both statements are true.
Key point
Points can be placed to form no triangle (minimum zero), and the maximum is C(6,3) = 20 triangles.
Remember
Six movable points: minimum 0 triangles, maximum C(6,3) = 20.
Common mistakes
- Assuming points on parallel lines always form triangles.
- Missing that points can coincide on their line.
- Counting only triples from different lines.
Exam tip
Maximum triangles = C(n, 3) when no three points are collinear; minimum can be zero if points coincide or are collinear.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)