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### Counting Angles and Rays

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Date: Wed, 23 Aug 1995 23:21:22 -0400
From: Anonymous
Subject: Angles and rays

My daughter asked me for help with this problem.

What is the algebraic expression to show the relation of the number
of angles created as rays are added from a common point?

For example--two rays create one angle
three rays create three angles
four rays create six angles

Is there a formula to know how many angles are created from 37 rays
from a common point??

Thanks in advance
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Date: 8/23/95 at 23:34:10
From: Doctor Ken
Subject: Re: Angles and rays

Hello!

To do this problem, you have to ask yourself "what is it that is really
defining the angle here?"  The angle is being defined by two of the rays
coming from the common point.  So to figure out how many angles are being
formed, you have to figure out how many ways you can choose two objects from
a collection of 37 objects, when order isn't important.

Well, it just so happens that there's a really handy formula that can tell
us exactly that.  It's called the "choose" formula.  It says that the number
of ways of choosing p objects from a collection of n objects, when order
doesn't matter (choosing one then another is the same as choosing them in the
reverse order) is n!/(p!(n-p)!).  I'll write it here as C[n,p].  So in our
case, we're choosing 2 rays from a collection of n rays, so our formula
becomes n!/(2(n-2)!) = n(n-1)/2.  In the case of 37 rays, that's 666 (ooh)
rays.

-Doctor Ken,  The Geometry Forum
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Associated Topics:
Middle School Geometry
Middle School Two-Dimensional Geometry

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