document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=476358>Question 476358</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear math teacher,\r<BR>\n" );
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document.write( "I am having difficulties with the following problem: <BR>\n" );
document.write( "How many straight lines are determined by n points, no three of which lie in the same straight line? \r<BR>\n" );
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document.write( "Here is how I reasoned through the problem:\r<BR>\n" );
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document.write( "n = n<BR>\n" );
document.write( "n = total number of points <BR>\n" );
document.write( "3 =  points that are NOT collinear<BR>\n" );
document.write( "(n-3) = points that ARE collinear\r<BR>\n" );
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document.write( "\"no three of which lie in the same straight line\" means no three of n points lie in the same straight line; therefore,<BR>\n" );
document.write( "A.) 3 points are NOT collinear and make nC3 =  n!/((n-3)!3!)lines = n(n-1)(n-2)(n-3)!/(n-3)!3! = n(n-1)(n-2)/3! = n(n^2-3n+2)/3! = n^3-3n^2+2n/3! lines<BR>\n" );
document.write( "B.) (n-3) points - are collinear points and make 1 line only.  They are also points remaining from n points after the 3 points are selected in nC3 ways as in A.)  \r<BR>\n" );
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document.write( "Total number of lines created from n points, no three of which lie in the same straight line = 1 line + n(n-1)(n-2)/3!; however, the textbook's answer is n(n-1)/2 lines. \r<BR>\n" );
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document.write( "Would you please correct me in this problem and explain to me what does \"no three of which lie in the same straight line\" truly mean? \r<BR>\n" );
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document.write( "Thank you very much for helping me figure this out. \r<BR>\n" );
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document.write( "Yours respectfully, \r<BR>\n" );
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document.write( "Ivanka<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326724 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=326724\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Two points determine a unique line. You choose any two of these n points (order does not matter), so the number of lines is nC2 = n(n-1)/2. The phrase \"no three which are collinear\" means that no three points lie on the same line; this is good because you do not have to worry about overcounting lines. </SPAN>\n" );
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