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" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326631 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=326631\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( " \"no three of which lie in the same straight line\" truly mean that a line is defined by two points;all you need is two points to draw a line through them. Through three points you can draw three lines (imagine triangle, extend its sides to make lines)\r<BR>\n" );
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document.write( "here are steps how to get a formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is your answer\r<BR>\n" );
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document.write( "1.<BR>\n" );
document.write( "Draw, or suppose you have, n points in a plane. No three points lie in a straight line. You want to know how many lines can be drawn through two points at a time.\r<BR>\n" );
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document.write( "For example, you may have a circle with eight points, denoted A through H.\r<BR>\n" );
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document.write( "2.<BR>\n" );
document.write( "Pick one point and determine how many pairs of points it can be in. If there are n points, the answer is n-1. This is how many lines can pass through that first point and another point at the same time.\r<BR>\n" );
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document.write( "Continuing with the above example, A can be matched up with B or C or D or E or F or G or H. That's seven possible matches.\r<BR>\n" );
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document.write( "3.\r<BR>\n" );
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document.write( "Pick the next point over. Its pairing with the first point has already been counted, but its pairing with the n-2 other points hasn't. Add n-2 to your earlier number, n-1, as possible lines through the points.\r<BR>\n" );
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document.write( "Continuing with the above example, B can have a line going through it and C through H. You don't count a line going through B and A, since you already did that in Step 2. So the possible lines through B are six.\r<BR>\n" );
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document.write( "4.\r<BR>\n" );
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document.write( "Continue with the pattern, adding n-3, then n-4, and so on. So the total sum of possible lines is n-1 + n-2 + n-3 + ... + 1. This is the same as summing up 1 + 2 + 3 + ... + n-1. It can be shown that the formula for 1 + 2 + 3 + ... + n-1 is:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Continuing with the above example, there were eight points, so n=8 gives a total number of possible lines through the points of n(n-1)/2 = 8*7/2 = 28. You can verify this yourself by adding the 7 found in Step 2 to the 6 found in Step 3 to 5, 4, 3, 2 and 1 to get 28. It also matches the result discussed in the introduction where the number of points was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2+=+3%2A2%2F2+=+3 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2+=+3%2A2%2F2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible lines.\r<BR>\n" );
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document.write( "PS:\r<BR>\n" );
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document.write( "nadam se da je sada jasno Ivanka...:-)\r<BR>\n" );
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