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polygon diagonals

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by pappueshwar » Tue Mar 13, 2012 9:20 am
There is an 8-polygons. what is the number of the diagonals of the 8-polygons ?
A. 20 B. 27 C. 32 D. 43 E. 45


is there any formula for this ?
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by tpr-becky » Tue Mar 13, 2012 11:39 am
when it asks "how many different" they are normally asking for permutations combinations and here that is the case however there is a difference.

First you would solve for the number of combinations of two points out of 8 possible points.
8!/(8-2)!(2!) = 28

However, this accounted for the sides, as a connection between two points that are right next to each other on the shape will form a side, not a diagonal. Therefore you have to subtract the 8 sides from this calculation to get 20

The correct answer is A.

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by pappueshwar » Tue Mar 13, 2012 7:01 pm
thanks that was awesome
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by Brent@GMATPrepNow » Thu Mar 15, 2012 8:52 am
pappueshwar wrote:There is an 8-polygons. what is the number of the diagonals of the 8-polygons ?
A. 20 B. 27 C. 32 D. 43 E. 45

is there any formula for this ?
Another approach is to recognize that, for each of the 8 points, there are 5 possible points that we can connect to to create a diagonal.

Aside: there are 5 possible points because we cannot create a diagonal by connecting 2 adjacent points.

These means that there are 40 possible diagonals (8x5=40).
However, we need to recognize that this method counts each diagonal twice. For example, it counts the diagonal AB and the diagonal BA as 2 separate diagonals.

So, to account for this duplication, we'll divide 40 by 2 to get 20[spoiler] (A)[/spoiler]

Cheers,
Brent
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