document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1160657>Question 1160657</A>: <I><SPAN STYLE=\"word-break: break-all;\"> While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #784119 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=784119\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I tried a few ways to analyze the problem using combinatorics and counting techniques, but it always got very messy.  So I decided the only way I was going to find the answer was by enumerating all the possibilities.<br><BR>\n" );
document.write( "Note that there might be an easy combinatorical path to the answer; but I didn't see it.<br><BR>\n" );
document.write( "For ease of analysis, instead of considering arrangements of ABCDEF with no two consecutive letters next to each other, I modeled the problem as arrangements of 123456 with no two consecutive digits next to each other.<br><BR>\n" );
document.write( "I will describe a bit of the analysis I did and let you have the pleasure of doing the same kind of logical analysis to finish the problem.<br><BR>\n" );
document.write( "Suppose, for example, that the numbers 2 and 4 are first (i.e., Billy and Dahlia are in the first two seats).  Numbers 1, 3, 5, and 6 are left; but the next number can't be either 3 or 5.  So if 2 and 4 are the first two, the first three are either 2-4-1 or 2-4-6.<br><BR>\n" );
document.write( "If the first three are 2-4-1, then 3, 5, and 6 are left.  The 5 and 6 have to be separated, so we have two allowable seating arrangements: 2-4-1-5-3-6 and 2-4-1-6-3-5.<vr><BR>\n" );
document.write( "If the first three are 2-4-6, then 1, 3, and 5 are left.  The 5 can't be next; either the 1 or the 3 can be next, and then the others can be in either order.  This case gives us 4 allowable seating arrangements: 2-4-6-1-3-5, 2-4-6-1-5-3, 2-4-6-3-1-5, and 2-4-6-3-5-1.<br><BR>\n" );
document.write( "Perform a similar analysis for all possible pairs of occupants of the first two seats.<br><BR>\n" );
document.write( "As careful as I tried to be, I found it aggravatingly easy to overlook some allowable arrangements.  As a check on the final list you end up with, note that, for any allowable arrangement, the mirror image arrangement is also allowable -- so the allowable arrangements will occur in pairs.  For example, the arrangements 2-4-1-5-3-6 and 2-4-1-6-3-5 will be matched with the arrangements 6-3-5-1-4-2 and 5-3-6-1-4-2.<br><BR>\n" );
document.write( "I found a total of 90 allowable arrangements....<br><BR>\n" );
document.write( "Have fun....!<br><BR>\n" );
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