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)heptagon

b0octagon

c)nonagon

d)decagon

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.

i hope this answer will help you.