There are 10 points in a plane. Exactly 4 of them lie on one straight line, and apart from these 4, no three of the points lie on a straight line. How many triangles can be formed with corners at these points?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 116
Any 3 points give a triangle unless they lie on one line; such choices are taken away from all the choices of 3 points.
All choices of 3 points from 10: \(\displaystyle ^{10}C_3=\frac{10\times9\times8}{6}=120\)
Choices of 3 points from the 4 points on one line (no triangle): \(^4C_3=4\)
Triangles \(=120-4=116\)
Why the other options are incorrect
Option (1) 120 — Incorrect. This counts the 4 choices of 3 points on the line, which give no triangle.
Option (2) 114 — Incorrect. This takes away \(^4C_2=6\); but a triangle is lost only when all 3 points are on the line, that is \(^4C_3=4\) cases.
Option (4) 96 — Incorrect. This takes away \(^4P_3=24\), which counts each set of 3 points on the line 6 times.