The number of triangles that are formed by choosing the vertices from a set of 12 points,

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 = 12C−  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.

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