document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199761>Question 1199761</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i) In how many ways six men and four women sit in a row<BR>\n" );
document.write( "ii) In how many ways can they sit in a row if all the men sit together<BR>\n" );
document.write( "iii) In how many ways can they sit in a row if just the women sit together<BR>\n" );
document.write( "iv) In how many ways can they sit in a row if men sit together\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833732 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=833732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (i)\r<BR>\n" );
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document.write( "6 men + 4 women = 10 people total\r<BR>\n" );
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document.write( "10 ways to fill the first slot<BR>\n" );
document.write( "9 to fill the second slot<BR>\n" );
document.write( "8 to fill the third slot<BR>\n" );
document.write( "and so on\r<BR>\n" );
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document.write( "10*9*8*7*6*5*4*3*2*1 = 10! = 3,628,800\r<BR>\n" );
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document.write( "The exclamation mark indicates factorial.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>3,628,800</font><BR>\n" );
document.write( "This value is roughly 3.63 million.\r<BR>\n" );
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document.write( "Part (ii)\r<BR>\n" );
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document.write( "Let's say the four women have code names A,B,C,D<BR>\n" );
document.write( "We'll have the men temporarily leave the group.\r<BR>\n" );
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document.write( "Another woman with code name \"E\" will be introduced. <BR>\n" );
document.write( "Wherever woman E is located, she represents the block of all men sitting together.\r<BR>\n" );
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document.write( "We have five letters A,B,C,D,E<BR>\n" );
document.write( "with 5*4*3*2*1 = 5! = 120 permutations\r<BR>\n" );
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document.write( "Within any of these 120 permutations, there are 6! = 720 ways to rearrange the men.\r<BR>\n" );
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document.write( "Therefore, we have 5!*6! = 120*720 = 86,400 different ways to arrange the people such that the men stay together.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>86,400</font>\r<BR>\n" );
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document.write( "Part (iii)\r<BR>\n" );
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document.write( "We'll follow very similar logic as compared to the previous part.<BR>\n" );
document.write( "Let's say the six men have code names A,B,C,D,E,F\r<BR>\n" );
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document.write( "The women leave the group temporarily, and in their place a man with code name \"G\" steps in. <BR>\n" );
document.write( "Wherever man G is, the block of women will step in his place.\r<BR>\n" );
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document.write( "There are 7 letters in A,B,C,D,E,F,G with 7! = 5040 permutations.\r<BR>\n" );
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document.write( "Within each of those permutations, there are 4! = 24 ways to arrange the women in their block.\r<BR>\n" );
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document.write( "7!*4! = 5040*24 = 120,960 total ways to arrange the people if all the women stay together. This time the men can separate.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>120,960</font>\r<BR>\n" );
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document.write( "Part (iv)\r<BR>\n" );
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document.write( "This part seems like a repeat of part (ii). <BR>\n" );
document.write( "Please double-check to make sure you typed the question correctly.\r<BR>\n" );
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document.write( "If there isn't a typo, then this question has very similar wording to part (ii).<BR>\n" );
document.write( "The key omission here is the word \"all\".\r<BR>\n" );
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document.write( "I interpret this to mean that some of the men can sit together, while others do not. Perhaps we have 2 or more blocks of men. Or we have 1 pairing of men, and the rest of the men are separate islands so to speak.\r<BR>\n" );
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document.write( "M = man<BR>\n" );
document.write( "W = woman\r<BR>\n" );
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document.write( "A sequence like M,W,M,M has one man on his own island, while the last two men are grouped together.<BR>\n" );
document.write( "The two sets of men are separated by a woman in the second slot.\r<BR>\n" );
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document.write( "Let's try to construct a scenario in which all of the men are separated. In other words: we'll try to form 6 separate islands of men.\r<BR>\n" );
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document.write( "There are 4 women. This gives the sequence<BR>\n" );
document.write( "_, W, _, W, _, W, _, W, _\r<BR>\n" );
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document.write( "The blanks represent a potential spot for a man.<BR>\n" );
document.write( "We have 4 women and 4+1 = blank spots.\r<BR>\n" );
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document.write( "We update to<BR>\n" );
document.write( "M, W, M, W, M, W, M, W, M<BR>\n" );
document.write( "The 6th man will go somewhere in this sequence. \r<BR>\n" );
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document.write( "No matter how hard we try, there's no way to have this 6th man get his own island. He'll be grouped with one of the initial 5 men already in the sequence.\r<BR>\n" );
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document.write( "What to take away from this: It's impossible to have the 6 men be fully separated. Meaning that each of the 10! = 3,628,800 different permutations will involve at least a pairing of two men.\r<BR>\n" );
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document.write( "Once again, the wording for part (iv) is similar to the wording in part (ii); which makes me think there might be a typo. <BR>\n" );
document.write( "Or the wording seems a bit vague.<BR>\n" );
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