document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=621807>Question 621807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let n>4.\r<BR>\n" );
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document.write( "In how many ways can we choose 4 vertices of a convex n-gon so as to form a convex quadrilateral, such that at least 2 sides of the quadrilateral are sides of the n-gon?<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 #391011 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=391011\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Label the vertices <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_1A_2...A_n\">.\r<BR>\n" );
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document.write( "We have two cases:\r<BR>\n" );
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document.write( "Case 1: Three of the four sides of the quadrilateral are sides of the n-gon.\r<BR>\n" );
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document.write( "Case 2: Exactly two of the four sides are sides on the n-gon.\r<BR>\n" );
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document.write( "Case 1 is pretty easy to count; all four vertices have to be consecutive. Hence we can have quadrilaterals <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_1A_2A_3A_4\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_2A_3A_4A_5\">, ..., <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_nA_1A_2A_3\">, n quadrilaterals.\r<BR>\n" );
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document.write( "Case 2 is a little trickier. The n-gon obviously has n-sides, so the solution is to choose two sides on the n-gon that are not consecutive. This can occur in\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{n(n-3)}{2}\"> ways (we must divide by 2 because we are counting each pair of sides twice).\r<BR>\n" );
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document.write( "Hence the total number of ways to choose four vertices in this manner is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n + \frac{n(n-3)}{2} = \frac{n^2 - n}{2}\">, which happens to be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE {{n}\choose{2}}\">. </SPAN>\n" );
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