Nine points are marked on a circle. How many different triangles can be drawn using these points as corners?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 84
A triangle is fixed by any 3 corners. No three points of a circle lie on one straight line, so every choice of 3 points gives a triangle: the count is \(^nC_3\).
Points \(n=9\)
Triangles \(\displaystyle =\,^9C_3=\frac{9\times8\times7}{3\times2\times1}=\frac{504}{6}=84\)
Why the other options are incorrect
Option (1) 504 — Incorrect. This is \(^9P_3\), which counts the same triangle 6 times, once for each order of its corners.
Option (3) 27 — Incorrect. This is \(9\times3\), which is not a counting rule.
Option (4) 36 — Incorrect. This is \(^9C_2\), the number of lines joining two points, not triangles.