MBA2010HereWeGo wrote:How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?
21
170
340
357
420
I am confused...pls help!
An n-sided convex polygon has nC2 different line segments that could possibly be drawn, out of which n are the sides of the polygon, and hence the total number of diagonals can be given by
nC2 - n = [n (n - 1)/2] - n = n (n - 3)/2
AB and BA are two different representations of the same line segment, call it AB or BA; the division by 2 is hence there in the resulting formula. If it's known that each of the n vertices has exactly (n - 3) one-way-read diagonals to its name, then the n vertices would have a total of n (n - 3) two-way-read diagonals or just n (n - 3)/2 diagonals to name. When one vertex does not participate in the diagonal formation, its share of exactly (n - 3) one-way-read diagonals is out from the total, and the remaining number of diagonals can be given by
[n (n - 3)/2] - (n - 3) = (n - 2) (n - 3)/2.
We have, n = 21, so our answer must be (21 - 2) (21 - 3)/2 = [spoiler]
171, oopsy!![/spoiler]
