document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=134264>Question 134264</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 15 points are arranged on a circle, how many lines are needed to join every point to every other point once?\r<BR>\n" );
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document.write( "Please help me!!\r<BR>\n" );
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document.write( "- Jordan </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #98190 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=98190\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One way to start thinking about it is to play around with a fewer number of<BR>\n" );
document.write( "points, say 5 equally spaced around a circle. Label the points 1-5<BR>\n" );
document.write( "Starting at point 1, draw lines to the other 4 points. You have 4 lines,<BR>\n" );
document.write( "or (5 - 1). <BR>\n" );
document.write( "From point 2, draw lines to the other 4 points. You have 4 lines, but the <BR>\n" );
document.write( "line to point 1 was already there, so subtract 1.<BR>\n" );
document.write( "So far you have<BR>\n" );
document.write( "from point 1: (5 - 1)<BR>\n" );
document.write( "from point 2: (5 - 1) - 1<BR>\n" );
document.write( "From point 3 draw lines to the other 4 points. You have 4 lines, but the<BR>\n" );
document.write( "lines to point 1 and point 2 were already there, so subtract 2<BR>\n" );
document.write( "Now finish for all 5 points<BR>\n" );
document.write( "from point 1: (5 - 1)<BR>\n" );
document.write( "from point 2: (5 - 1) - 1<BR>\n" );
document.write( "from point 3: (5 - 1) - 2<BR>\n" );
document.write( "from point 4: (5 - 1) - 3<BR>\n" );
document.write( "from point 5: (5 - 1) - 4<BR>\n" );
document.write( "Doing the additions and subtractions , I get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A4+-+10+=+10 ALIGN=MIDDLE  ALT=\"5%2A4+-+10+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > lines. I counted them- it works<BR>\n" );
document.write( "Now I can come up with a formula for any number of points<BR>\n" );
document.write( "arranged on a circle and use it for 15 points<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > = the number of points <BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > = the number of different lines connecting the points<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+n%2A%28n+-+1%29 ALIGN=MIDDLE  ALT=\"k+=+n%2A%28n+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > - (the sum 1 + 2 + 3 + 4 + . . . + (n - 1))<BR>\n" );
document.write( "There is a formula for the sum. It is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n-1%29%2F2%29%28%28n-1%29%2B1%29 ALIGN=MIDDLE  ALT=\"%28%28n-1%29%2F2%29%28%28n-1%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2A%28n-1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28n%2A%28n-1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So far, the formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+n%2A%28n-1%29+-+n%2A%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+n%2A%28n-1%29+-+n%2A%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+n%2A%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+n%2A%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > general formula<BR>\n" );
document.write( "Now see if this formula works for 5 points<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+5%2A%285-1%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+5%2A%285-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+10 ALIGN=MIDDLE  ALT=\"k+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > It works<BR>\n" );
document.write( "Now for 15 points<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+15%2A14%2F2 ALIGN=MIDDLE  ALT=\"k+=+15%2A14%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+105 ALIGN=MIDDLE  ALT=\"k+=+105\"   DISABLED_style= \"max-width:90%; height:auto;\" > answer<BR>\n" );
document.write( "Hope this didn't confuse you<BR>\n" );
document.write( "The general formula should work for any number of points<BR>\n" );
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