document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=344851>Question 344851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We are to seat 5 boys, 5 girls, and 1 parent in a circular arrangement. In how many ways can this be done if no boy is to sit next to a boy and no girl is to sit next to a girl? What if there are 2 parents?\r<BR>\n" );
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document.write( "I really just don't know where to start. I've drawn it up, but if you count symmetries and assume the boys and girls to be indistinguishable its weird.. i'm honestly just not good at this haha. thanks for all your help though  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #246776 by </B><A HREF=/tutors/aboutme.mpl?userid=scott8148><B>scott8148(6628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=scott8148><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=scott8148><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=246776\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the parent(s) are easy to factor in , just concentrate on the boys and girls\r<BR>\n" );
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document.write( "the only way for no two boys or two girls to be next to each other is for them to be alternating around the circle\r<BR>\n" );
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document.write( "imagine a ring of 5 boys (with spaces) and a ring of 5 girls (with spaces)<BR>\n" );
document.write( "___ the rings are intermeshed to make the seating arrangement\r<BR>\n" );
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document.write( "how many ways can five things be ordered or arranged sequentially?<BR>\n" );
document.write( "___ usually , 5!<BR>\n" );
document.write( "___ but a circular arrangement repeats itself as many times as there are positions on the circle<BR>\n" );
document.write( "___ so there are 4! (5! / 5) orders for the ring of boys and for the ring of girls<BR>\n" );
document.write( "___ there are also 5 ways for the rings to intermesh\r<BR>\n" );
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document.write( "so the number of boy-girl arrangements is ___ 4! * 4! * 5\r<BR>\n" );
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document.write( "one parent can insert a chair at any of 10 places on the circle<BR>\n" );
document.write( "___ so boys and girls and one parent is ___ 4! * 4! * 5 * 10\r<BR>\n" );
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document.write( "the second parent would have 11 places to insert a chair<BR>\n" );
document.write( "___ so boys and girls and two parents is ___ 4! * 4! * 5 * 10 * 11\r<BR>\n" );
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document.write( "the second parent would also allow an adjacent boy and girl to swap positions <BR>\n" );
document.write( "so that two boys and two girls<BR>\n" );
document.write( "would be separated by parents instead of other boys and girls<BR>\n" );
document.write( "___ 10 possible swaps with two positions for the parents<BR>\n" );
document.write( "___ the possible arrangements now becomes ___ 4! * 4! * 5 * (10 * 11 + 20) </SPAN>\n" );
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