Find the number of triangles formed by 12 poi | vedclass

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Question

(A) To form a triangle,we need $3$ non-collinear points.Total number of ways to select $3$ points from $12$ points is given by ${}^{12}C_{3}$.${}^{12}

Vedclass · Accepted Answer

$216$

Vedclass · Answer

(A) To form a triangle,we need $3$ non-collinear points.Total number of ways to select $3$ points from $12$ points is given by ${}^{12}C_{3}$.${}^{12}C_{3} = \frac{12 	imes 11 	imes 10}{3 	imes 2 	imes 1} = 220$.Since $4$ points are collinear,selecting any $3$ points from these $4$ will not form a triangle.The number of ways to select $3$ points from these $4$ collinear points is ${}^{4}C_{3} = 4$.Therefore,the total number of triangles formed = ${}^{12}C_{3} - {}^{4}C_{3} = 220 - 4 = 216$.

Find the number of triangles formed by $12$ points (out of which $4$ are collinear) in a plane.

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