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# How many circles can be drawn on the line segment

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Joined: 25 Nov 2011
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How many circles can be drawn on the line segment [#permalink]  25 Feb 2012, 23:05
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Difficulty:

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Question Stats:

79% (01:42) correct 21% (00:58) wrong based on 64 sessions
Assume that there are 12 points on a straight line as shown below:

--*-*-*----*-*-*-*-*--*-*-*-*

How many circles can be drawn on the line segment above if each circle is subject to the rule that the endpoints of one of its diameters must be any two of the points?

36
66
72
78
121
[Reveal] Spoiler: OA

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-Aravind Chembeti

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Re: How many circles can be drawn on the line segment [#permalink]  25 Feb 2012, 23:15
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Chembeti wrote:
Assume that there are 12 points on a straight line as shown below:

--*-*-*----*-*-*-*-*--*-*-*-*

How many circles can be drawn on the line segment above if each circle is subject to the rule that the endpoints of one of its diameters must be any two of the points?

A. 36
B. 66
C. 72
D. 78
E. 121

Any 2 different points out of given 12 colinear points can be the endpoints of the diameter, so there can be $$C^2_{12}=66$$ circles drawn.

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Re: How many circles can be drawn on the line segment [#permalink]  23 Oct 2013, 12:05
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Re: How many circles can be drawn on the line segment [#permalink]  23 Oct 2013, 18:10
Expert's post
This is an interesting way to disguise a combination problem. As Bunuel noted, you can create a circle through any two points, so what you really have is a question of how many unique ways to select pairs from a set of 12 points, or:

12!/2!(12-2)! --> 12!/2!10! --> (12*11)/2 --> 6*11 = 66
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Re: How many circles can be drawn on the line segment   [#permalink] 23 Oct 2013, 18:10
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