The number of triangles that can be formed by | vedclass

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(A) To form a triangle,we need to select $3$ non-collinear points from the given set of points.Total number of ways to select $3$ points from $12$ poi

Vedclass · Accepted Answer

$185$

Vedclass · Answer

(A) To form a triangle,we need to select $3$ non-collinear points from the given set of 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 $7$ points are collinear,any selection of $3$ points from these $7$ points will not form a triangle.The number of ways to select $3$ points from these $7$ collinear points is $^{7}C_3$.$^{7}C_3 = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = 35$.Therefore,the number of triangles that can be formed is the total number of selections minus the number of collinear selections:Number of triangles $= ^{12}C_3 - ^{7}C_3 = 220 - 35 = 185$.

The number of triangles that can be formed by choosing the vertices from a set of $12$ points,$7$ of which lie on the same straight line,is

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