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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 ______.
Options
105
15
175
185
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Solution
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 185.
Explanation:
Total number of triangles formed from 12 points taking 3 at a time = 12C3
But given that out of 12 points, 7 are collinear
So, these seven points will form no triangle.
∴ The required number of triangles = 12C3 – 7C3
= `(12!)/(3! 9!) - (7!)/(3!4!)`
= `(12 xx 11 xx 10 xx 9!)/(3 xx 2 xx 1 xx 9!) - (7 xx 6 xx 5 xx 4!)/(3 xx 2 xx 1 xx 4!)`
= `(12 xx 11 xx 10)/(3 xx 2) - (7 xx 6 xx 5)/(3 xx 2)`
= 220 – 35
= 185
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