document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202348>Question 1202348</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The prairie dogs were having a town meeting to talk about the blackfooted ferret that was terrorizing the neighborhood. One town council dog suggested building a network of tunnels connecting every burrow to every other burrow. Then someone said Are you crazy ? We have nine burrows. Thats too many tunnels nd too much money to build them. If each tunnel cost $1.00 to build, how much would it cost the prairie dogs to connect every burrow with every other burrow? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837116 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=837116\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>36 tunnels</font><BR>\n" );
document.write( "Total cost = $36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "====================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "\n" );
document.write( "Let's say for example there are 5 burrows labeled A,B,C,D,E.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Make a 5 by 5 table with those labels along the left and top side.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >A</TD><TD >B</TD><TD >C</TD><TD >D</TD><TD >E</TD></tr><tr><TD >A</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >B</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >C</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >D</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >E</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It's clear it makes no sense to build a tunnel connecting any given burrow to itself. We cannot connect A to A for instance.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's cross those entries out with an X.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >A</TD><TD >B</TD><TD >C</TD><TD >D</TD><TD >E</TD></tr><tr><TD >A</TD><TD >X</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >B</TD><TD ></TD><TD >X</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >C</TD><TD ></TD><TD ></TD><TD >X</TD><TD ></TD><TD ></TD></tr><tr><TD >D</TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD><TD ></TD></tr><tr><TD >E</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "The 5*5 = 25 entries has now dropped to 25-5 = 20 entries. We've subtracted off the 5 cells marked with X.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words:<BR>\n" );
document.write( "25 - 5<BR>\n" );
document.write( "5*5 - 5*1<BR>\n" );
document.write( "5*(5-1)<BR>\n" );
document.write( "5*4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Meaning that if we had n burrows, then we'd have n^2 - n = n*(n-1) entries after removing the northwest diagonal.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But notice the symmetry of the table. The entries marked with Y represent connecting burrow A to C<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >A</TD><TD >B</TD><TD >C</TD><TD >D</TD><TD >E</TD></tr><tr><TD >A</TD><TD >X</TD><TD ></TD><TD >Y</TD><TD ></TD><TD ></TD></tr><tr><TD >B</TD><TD ></TD><TD >X</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >C</TD><TD >Y</TD><TD ></TD><TD >X</TD><TD ></TD><TD ></TD></tr><tr><TD >D</TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD><TD ></TD></tr><tr><TD >E</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "We've double-counted things when marking two instances of Y.<BR>\n" );
document.write( "So we ignore either the lower half or the upper half. The string of X's is the divider line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means we'll divide by 2 when ignoring half of the cells of the table.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 5*4/2 = 20/2 = 10 different tunnels connecting the five burrows together.<BR>\n" );
document.write( "The ten tunnels are: <BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "AB\r\n" );
document.write( "AC\r\n" );
document.write( "AD\r\n" );
document.write( "AE\r\n" );
document.write( "\r\n" );
document.write( "BC\r\n" );
document.write( "BD\r\n" );
document.write( "BE\r\n" );
document.write( "\r\n" );
document.write( "CD\r\n" );
document.write( "CE\r\n" );
document.write( "\r\n" );
document.write( "DE\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "A tunnel like AB is the same as BA. Order doesn't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The formula in general is n*(n-1)/2 which is found in the nCr formula when r = 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we had n = 9 burrows, then,<BR>\n" );
document.write( "n*(n-1)/2<BR>\n" );
document.write( "9*(9-1)/2<BR>\n" );
document.write( "9*8/2<BR>\n" );
document.write( "72/2<BR>\n" );
document.write( "36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are <font color=red>36 tunnels</font> connecting the 9 burrows to each other. Each tunnel costs $1.00, which means the total cost is $36.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );