Question:
The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is
(a) 105
(b) 15
(c) 175
(d) 185
Solution:
Number of ways of selecting 3 points from given 12 points = 12C3 .
But any three points selected from given seven collinear points does not from triangle number ways of selecting three points team seven collinear points
= 7C3
∴ Required number of triangle = 12C3 − 7C3
$=\frac{12 !}{3 ! 9 !}-\frac{7 !}{3 ! 4 !}$
$=\frac{12 \times 11 \times 10}{3 \times 2}-\frac{7 \times 6 \times 5}{2 \times 3}$
$=220-35$
$=185$
Hence, the correct answer is option D.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.