document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=33110>Question 33110</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Four married couples have reserved 8 seats in a row at the theatre, starting at an aisle seat. How many ways can they arrange themselves if all the women sit together and all the men sit together? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #19530 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><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=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=19530\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I do not believe there is a quick \"put numbers into a formula\" approach to this. You need to think about it. Use a few sketches:\r<BR>\n" );
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document.write( "MMMMwwww or wwwwMMMM are the only 2 possible variations of the sets of men and women. The next thing we need to figure out is the number of ways to organise the 4 men and 4 women.\r<BR>\n" );
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document.write( "For the 4 men in MMMMwwww seating plan, there are the following number of permutations: 4! --> 24. This is gotten from the following process:\r<BR>\n" );
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document.write( "4 men. How many can sit in first seat? Answer 4.<BR>\n" );
document.write( "3 remaining men. How many can sit in the second seat? Answer 3. <BR>\n" );
document.write( "--> Ongoing total so far: 4*3 different permutations.<BR>\n" );
document.write( "2 remaining men. How many can sit in the third seat? Answer 2.<BR>\n" );
document.write( "--> Ongoing total so far: 4*3*2 different permutations.<BR>\n" );
document.write( "1 remaining man. How many can sit in the fourth position? Answer 1.<BR>\n" );
document.write( "--> Ongoing total so far: 4*3*2*1 different permutations.\r<BR>\n" );
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document.write( "So... 24 different ways that the 4 men can sit down.\r<BR>\n" );
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document.write( "Now the women, being 4 of them have also got 24 different ways of sitting down in their 4 seats.\r<BR>\n" );
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document.write( "For each of the 24 permutations for the men, there are 24 ways for the women to sit. So, in total, there are 24*24 --> 576 different ways for all 8 to sit in MMMMwwww.\r<BR>\n" );
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document.write( "And now, do not forget that all this is repeated for wwwwMMMM, so we have a total of 576*2 --> 1152 different permutations.\r<BR>\n" );
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document.write( "Might well be a formula, but i can never remember how to apply it to questions, so i always work it out this \"from first principles\" way.\r<BR>\n" );
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document.write( "Actual working out would be something like:<BR>\n" );
document.write( "MMMMwwww and wwwwMMMM\r<BR>\n" );
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document.write( "For MMMMwwww:<BR>\n" );
document.write( "4 men have 4! permutations --> 24<BR>\n" );
document.write( "4 women have 24 too.<BR>\n" );
document.write( "Together that makes 24*24 --> 576 permutations.\r<BR>\n" );
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document.write( "wwwwMMMM has the same number.<BR>\n" );
document.write( "Hence total is 576+576 --> 1152 permutations\r<BR>\n" );
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document.write( "jon.<BR>\n" );
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