In a plane, there are 9 points, out of which 4 are collinear. The number of triangles made by these points is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 80
A triangle needs three points that are not on one line. So the number of triangles is the number of ways of choosing 3 of the 9 points, less the choices of 3 points that all lie on the line of collinear points.
Ways of choosing 3 of the 9 points: \(\displaystyle {}^{9}C_{3}=\frac{9\times8\times7}{3\times2\times1}=84\)
Choices of 3 of the 4 collinear points, which give no triangle: \({}^{4}C_{3}=4\)
Number of triangles \(=84-4=80\)
The value 84 in Option (2) counts every choice of 3 points, including the 4 that lie on one line.