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# How many diagonals does a polygon with 21 sides have, if one

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Senior Manager
Joined: 02 Dec 2007
Posts: 419
How many diagonals does a polygon with 21 sides have, if one  [#permalink]

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27 Aug 2008, 08:33
How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

a) 21
b) 170
c) 340
d) 357
e) 420

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Manager
Joined: 15 Jul 2008
Posts: 205

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27 Aug 2008, 09:18
Nihit wrote:
How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

a) 21
b) 170
c) 340
d) 357
e) 420

I think 170...

take any point. You have 20 other points. Of these 20, 2 will be adjacent and so no diagonal with them. 18 possibilities. But one point has no diagonal with any other point. So for the chosen point, there are 17 diagonals possible.
This applies to 20 points (excluding the one that does not have diagonal with any point)

So you have 17*20. This includes double counting. So divide by 2

170.

I am interested in knowing how combination can be used to tackle this problem.
SVP
Joined: 07 Nov 2007
Posts: 1728
Location: New York

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27 Aug 2008, 09:33
bhushangiri wrote:
Nihit wrote:
How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

a) 21
b) 170
c) 340
d) 357
e) 420

I think 170...

take any point. You have 20 other points. Of these 20, 2 will be adjacent and so no diagonal with them. 18 possibilities. But one point has no diagonal with any other point. So for the chosen point, there are 17 diagonals possible.
This applies to 20 points (excluding the one that does not have diagonal with any point)

So you have 17*20. This includes double counting. So divide by 2

170.

I am interested in knowing how combination can be used to tackle this problem.

Let say polygon has n points.
chose any 2 points to make line.
= nC2 combinations (include all sides and all diagnols)
= n(n-1)/2
no of diagnols = n(n-1)/2 - n = n(n-3)/2

Here one vertex is not participating..
visualize that it is 20 sides polygon..
= 20*17/2 - 1 (side is actually not present ) +1 ( actually that side is diagnol)
=170
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Joined: 11 May 2008
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27 Aug 2008, 18:22
im getting 171. how come??
total poss diagonals = 21*18/2=189
but one vertex is not connected with any diagonal,
so we reduce the total no. by 21-3= 18 diagonals.
so 189-18=171.
?????

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This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Re: Diagonals &nbs [#permalink] 27 Aug 2008, 18:22
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# How many diagonals does a polygon with 21 sides have, if one

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