document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174094>Question 1174094</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In how many 7 children can be arranged in a line, such that two particular children of them are never together? <BR>\n" );
document.write( "a. 360<BR>\n" );
document.write( "b. 480<BR>\n" );
document.write( "c. 720<BR>\n" );
document.write( "d. 1440 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #799458 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=799458\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Label the chidren as A,B,C,D,E,F,G\r<BR>\n" );
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document.write( "Let's say we want child A and child B to never be sitting together. That's the ultimate goal that we'll aim for.\r<BR>\n" );
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document.write( "However, for now we'll consider the opposite and count the number of ways that we can arrange the children such that A and B are always together. <BR>\n" );
document.write( "What we'll do is remove both of these letters from the list, and replace them with H\r<BR>\n" );
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document.write( "So we go from<BR>\n" );
document.write( "A,B,C,D,E,F,G<BR>\n" );
document.write( "to<BR>\n" );
document.write( "H,C,D,E,F,G\r<BR>\n" );
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document.write( "H is a placeholder for either A,B or B,A.\r<BR>\n" );
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document.write( "There are 6 letters in the set {H,C,D,E,F,G} so there are 6! = 6*5*4*3*2*1 = 720 ways to permute these items.\r<BR>\n" );
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document.write( "Since there are two ways to replace H, for any of the 720 permutations mentioned, this means there are overall 2*720 = 1440 different ways to have the children arranged such that child A and child B are always sitting next to each other.\r<BR>\n" );
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document.write( "However, we want them separated. So we must subtract that count from 7! = 7*6*5*4*3*2*1 = 5040 because this represents the total number of ways to arrange all seven kids without any restrictions.\r<BR>\n" );
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document.write( "Phrased another way:<BR>\n" );
document.write( "answer we want = (number of ways to arrange all kids with no restrictions) - (number of ways to arrange the kids such that A,B are always together)\r<BR>\n" );
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document.write( "We could use algebra to tidy that equation up<BR>\n" );
document.write( "x = number of ways to arrange all kids with no restrictions (A and B may or may not be sitting together)<BR>\n" );
document.write( "y = number of ways to arrange the kids such that A,B are always together<BR>\n" );
document.write( "z = number of ways to arrange the kids where A,B are never together<BR>\n" );
document.write( "So based on that we can say,<BR>\n" );
document.write( "z = x-y\r<BR>\n" );
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document.write( "Earlier we found that<BR>\n" );
document.write( "x = 5040<BR>\n" );
document.write( "y = 1440<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "z = x-y<BR>\n" );
document.write( "z = 5040-1440<BR>\n" );
document.write( "z = 3600\r<BR>\n" );
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document.write( "Answer:  3600\r<BR>\n" );
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document.write( "I think your teacher made a typo and meant to say 3600 for choice A, instead of 360.<BR>\n" );
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