Find the no. of Diagonals of a Polygon of 15 sides....

Find the no. of Diagonals of a Polygon of 15 sides....


Sol  :

A polygon of n sides will have n vertices . A diagonal or a side of the polygon will be formed by joining any two vertices of the polygon..

No. of Diagonals of the polygon + No. of sides of the polygon = C(n,2) - n

                                                                             = n! / 2!(n-2)!  - n

                                                                             = n x (n-1) x (n-2)! / 2!(n-2)!  - n

                                                                             = n x (n-1) / 2   - n


                                                                             = {n(n-1) - 2m} / 2

                                                                             = n{n-1-2} / 2

                                                                             = n(n-3) / 2


Now,     Putting n = 15,  We get the number of Diagonals   = 15(15-3)  / 2

                                                                              = 15 x 12 / 2

                                                                              = 90