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three vertices of a convex  n sided polygon are selected .if the number of trianges that can be constructed such that none of the sides of the triangle is also the side of  the polygon is 30 . then the polygon is a)heptagonb0octagonc)nonagond)decagon

Yash Chourasiya
one year ago
Hello Student

Any combination of3 vertices can serve to form a triangle. Number of combinations of 3that can be formed from n choices =(nC3)

Each side can be combined with any NON-ADJACENT vertex to form a triangle with exactly 1 side in common with the polygon. Since there arensides, each withn−4 non-adjacent vertices,
the total number of ways =n(n−4).

Then
(nC3)−n(n−4)−n=30
n3−9n2+20n−180=0
On simplifying, we get
n = 9
means polygon is nonagon.