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

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How many diagonals does a polygon with 21 sides have, if one [#permalink]  06 Aug 2007, 18:32
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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
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I'm tempted to go with B.

# of diagonals a polygon with 21 side have = 21(21-3)/2 = 189
So if one vertex does not connect anywhere else, then the # of diagonals must be less than 189. This leaves only A and B. Choice A seems to be a little on the low side, so I'll take B.
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Here are my thoughts:

21 (# vertices) * 18 (# diagonals leaving from each vertex) / 2 (to avoid double counting) = 189 unique diagonals

If one vertex has no diagonals, the equation becomes
20 (# vertices) * 17 (# diagonals leaving from each vertex) / 2 (to avoid double counting) = 170 unique diagonals
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another way to look at this :

2 vertices out of 20 can be chosen in : 20 C 2 : 190

since this also include the sides of the polygon so for 20 of those you need to subtract 20 out of the 190.

Hence : 190 - 20 = 170
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