document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209093>Question 1209093</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. In how many ways can 3 male students and 2 female students be seated on 6 chairs arranged in a circle such that no two male students are adjacent?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2. In how many ways can 3 male students and 2 female students be seated on 7 chairs arranged in a circle in the following cases:\r<BR>\n" );
document.write( "\n" );
document.write( "1. The female students are adjacent.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2. The male students are adjacent.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3. The male students are adjacent, and the female students are adjacent.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4. No two male students are adjacent.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5. No two female students are adjacent.\r<BR>\n" );
document.write( "\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 #847669 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52852)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "1. In how many ways can 3 male students and 2 female students be seated on 6 chairs <BR>\n" );
document.write( "arranged in a circle such that no two male students are adjacent?<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In this my post, I will solve problem 1, only.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As this problem is posed (with one vacant chair), I may assume that <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;you are just familiar with more simple problems without vacant chair<BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and know the basics of placement around circular table, as well as<BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the basics of circular permutations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\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( "We have 6 chairs, in all.  Of them, one chair is vacant, 3 male students sit on their chairs \r\n" );
document.write( "such that no two male students are adjacent, and two female students sit on their chairs.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Having round table and circular permutations around it, we can assume that the vacant chair is \r\n" );
document.write( "in position \"North\", or at \"12 o'clock\" (position n1).  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then for 3 male students and two female students we have chairs in five remaining positions 2, 3, 4, 5, and 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Since no two the male students are adjacent, it means that three male students occupy 3 chairs with even numbers 2, 4, 6,\r\n" );
document.write( "while two female students occupy 2 chairs with odd numbers 3 and 5.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then for three male students we have 3! = 6 possible permutations on their chairs, and for two female students \r\n" );
document.write( "we have 2! = 2 possible permutations on their chairs.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In all, it gives us all 6*2 = 12 possible distinguishable circular arrangements\r\n" );
document.write( "satisfying the imposed restrictions.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "Solved.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );