document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207318>Question 1207318</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi can you help me with this problem. Does it matter what the couple are, or do<BR>\n" );
document.write( "we just say two people are a couple. <BR>\n" );
document.write( "The drama club buys tickets for 12 seats in a row for a local production, and<BR>\n" );
document.write( "then sends the tickets randomly to the 12 people who ordered seats. Of these 12<BR>\n" );
document.write( "people, 7 are female and 5 are male. Determine the probability that:<BR>\n" );
document.write( "a) A couple will receive tickets sitting together.<BR>\n" );
document.write( "b) The women will be seated together and the men will be seated together.<BR>\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845126 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=845126\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "(a) <font color=red>1/6</font><BR>\n" );
document.write( "(b) <font color=red>1/396</font>\r<BR>\n" );
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document.write( "Explanation for Part (a)\r<BR>\n" );
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document.write( "The first 12 letters of the English alphabet are A through L.\r<BR>\n" );
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document.write( "Let A and B represent the two people that make up this particular couple that want to sit together.\r<BR>\n" );
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document.write( "Let X represent their place in any given permutation. <BR>\n" );
document.write( "The sequence C,D,X,E,F,G,H,I,J,K,L is one such example.<BR>\n" );
document.write( "We'll replace X with either A,B or B,A.\r<BR>\n" );
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document.write( "Doing it this way guarantees that A & B stick together.<BR>\n" );
document.write( "i.e. it guarantees that A & B are next door neighbors.\r<BR>\n" );
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document.write( "There are initially 12 letters.<BR>\n" );
document.write( "Removing A & B temporarily drops things to 12-2 = 10 letters.<BR>\n" );
document.write( "Temporarily introducing X bumps it up to 10+1 = 11 letters.\r<BR>\n" );
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document.write( "There are 2*11! ways to arrange the 12 people so that A,B sit together in either order. <BR>\n" );
document.write( "The exclamation mark indicates factorial.<BR>\n" );
document.write( "The 11! factorial term is the number of ways to arrange the 11 letters involving X (but not A,B just yet), and the 2 out front is to consider the two cases when A,B happens or B,A happens.\r<BR>\n" );
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document.write( "This is out of 12! ways to arrange all 12 people regardless if that particular couple is sitting together or not.\r<BR>\n" );
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document.write( "So we have:<BR>\n" );
document.write( "2*11! = number of ways to arrange people so A,B stick together<BR>\n" );
document.write( "12! = number of permutations total\r<BR>\n" );
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document.write( "Divide the two items mentioned:<BR>\n" );
document.write( "(2*11!)/(12!)<BR>\n" );
document.write( "(2*11!)/(12*11!)<BR>\n" );
document.write( "2/12<BR>\n" );
document.write( "<font color=red>1/6</font> is the answer to part (a).<BR>\n" );
document.write( "Notice the 11! terms cancel which is really convenient.\r<BR>\n" );
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document.write( "Another thing to note is if you had n people then 2/n is the probability that one particular couple sticks together. <BR>\n" );
document.write( "The proof is short so I'll leave it to the reader. <BR>\n" );
document.write( "Hint: 2*(n-1)! is the number of ways to arrange n people so that exactly one couple sticks together. \r<BR>\n" );
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document.write( "I recommend trying small values of n to see examples of why this formula works. <BR>\n" );
document.write( "Here's a small bit of scratch work when n = 3<BR>\n" );
document.write( "The double star notation indicates when A,B are together<BR>\n" );
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document.write( "ABC **\r\n" );
document.write( "ACB\r\n" );
document.write( "BAC **\r\n" );
document.write( "BCA\r\n" );
document.write( "CAB **\r\n" );
document.write( "CBA **\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "There are 4 cases when A,B are together out of 6 permutations.<BR>\n" );
document.write( "4/6 = 2/3 is the probability of A,B being together in either order.<BR>\n" );
document.write( "This matches with the formula 2/n where n = 3. So this example helps confirm the formula works. I recommend trying it with n = 4 to see what you find. \r<BR>\n" );
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document.write( "A similar question is found here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://math.stackexchange.com/questions/114367/probability-people-sitting-in-a-row-linear-arrangement\">https://math.stackexchange.com/questions/114367/probability-people-sitting-in-a-row-linear-arrangement</a><BR>\n" );
document.write( "The \"solution\" 1/63 mentioned on that page is NOT correct. I'm not sure why the professor made such a strange error. Instead the solution is 1/5 as the other people point out.\r<BR>\n" );
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document.write( "Explanation for Part (b)\r<BR>\n" );
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document.write( "Let M and W represent the block of men and block of women respectively.<BR>\n" );
document.write( "We could have MW or WM<BR>\n" );
document.write( "So there are 2 ways to arrange these blocks.\r<BR>\n" );
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document.write( "Within any given such arrangement, we have 7! ways to arrange the 7 women within their respective block, and 5! ways to arrange the 5 men in their respective block.\r<BR>\n" );
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document.write( "2*7!*5! is the number of ways to arrange the people so that the genders are fully separated.<BR>\n" );
document.write( "This is out of 12! ways to arrange all 12 people regardless if the genders are blocked together or not.\r<BR>\n" );
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document.write( "Divide the items mentioned.<BR>\n" );
document.write( "(2*7!*5!)/(12!)<BR>\n" );
document.write( "(2*7!*5!)/(12*11*10*9*8*7!)<BR>\n" );
document.write( "(2*5!)/(12*11*10*9*8)<BR>\n" );
document.write( "(2*5*4*3*2*1)/(12*11*10*9*8)<BR>\n" );
document.write( "(4*3*2*1)/(6*11*2*9*8)<BR>\n" );
document.write( "(2*1)/(6*11*2*3*2)<BR>\n" );
document.write( "(1)/(6*11*2*3)<BR>\n" );
document.write( "1/(66*6)<BR>\n" );
document.write( "<font color=red>1/396</font> is the answer to part (b)<BR>\n" );
document.write( "The steps above hopefully show how I cancelled things. Other routes are possible.<BR>\n" );
document.write( "You can use a calculator to make quick work of this, but it still might be useful to know how to do this by hand. <BR>\n" );
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