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

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Director
Joined: 27 Jun 2005
Posts: 504
Location: MS
How many diagonals does a polygon with 21 sides have, if one [#permalink]

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09 Feb 2006, 18:00
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. 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
VP
Joined: 21 Sep 2003
Posts: 1057
Location: USA

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09 Feb 2006, 18:49
Somehow I am getting 171

All the lines that can be drawn with 21 points = 21C2 =
Number of sides = 21

So number of diagonals when diagonals are drawn from all edges = 210 - 21 = 189 (Subtract the number of sides)

Number of missing diagonals when one of the vertex does not have any diagonals= 21-3 = 18 (By drawing some of the known polygons... you can see n-3 disgonals will be missing)

Total = 189-18 = 171

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Senior Manager
Joined: 05 Jan 2006
Posts: 381

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10 Feb 2006, 01:23
21 points One does not participate...20 points.

So joining 20 points combinations are C(20,2) = 10*19 = 180

Outof this there will be 20-1=19 are valid sides of polygon so 180-19=171...
Senior Manager
Joined: 11 Nov 2005
Posts: 328
Location: London

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10 Feb 2006, 02:06
C. 340

polygon 0f 21 sides.... one side does not connect to any diagonal..

21-1= 20, from any point there will be 17 diagonal (20-3) because the point in consideration is connected to two other point of the polygon

17*20 = 340
Senior Manager
Joined: 05 Jan 2006
Posts: 381

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10 Feb 2006, 09:30
SunShine wrote:
C. 340

polygon 0f 21 sides.... one side does not connect to any diagonal..

21-1= 20, from any point there will be 17 diagonal (20-3) because the point in consideration is connected to two other point of the polygon

17*20 = 340

If A connects to B and B connects back to A both are same diagonal!
so 340/2 = 170

More over you also need to consider one special case where end two points conencts to each other so its 170+1=171 !

what is OA/OE?
Senior Manager
Joined: 11 Jan 2006
Posts: 269
Location: Chennai,India

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11 Feb 2006, 04:48
giddi77 wrote:
Somehow I am getting 171

All the lines that can be drawn with 21 points = 21C2 =
Number of sides = 21

So number of diagonals when diagonals are drawn from all edges = 210 - 21 = 189 (Subtract the number of sides)

Number of missing diagonals when one of the vertex does not have any diagonals= 21-3 = 18 (By drawing some of the known polygons... you can see n-3 disgonals will be missing)

Total = 189-18 = 171

Giri , i came up to the same approach till the last step where u subtracted 3 from 21 ! can u explain that for me.
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11 Feb 2006, 04:48
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