document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=265871>Question 265871</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question: I have 12 textbooks to arrange on a bookshelf. 5 of them are English textbooks and I would like to arrange the books so that at least two of the English textbooks are adjacent to one another. How many different arrangements are possible?\r<BR>\n" );
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document.write( "This is how far I have gotten:<BR>\n" );
document.write( "(# of ways that all are touching) - (# of ways that none are touching) <BR>\n" );
document.write( "(5!)(7!)(2!) - ????\r<BR>\n" );
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document.write( "If I am on the right track... how do I figure out the (# of ways that none are touching)?\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 #195454 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=195454\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Question: I have 12 textbooks to arrange on a bookshelf. 5 of them are English textbooks and I would like to arrange the books so that at least two of the English textbooks are adjacent to one another. How many different arrangements are possible?<BR>\n" );
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document.write( "Answer: total # of arrangements - # of arrangements with no two english adjacent<BR>\n" );
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document.write( "# no 2 are adjacent = ?<BR>\n" );
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document.write( "Four non-englsh will be required to separate the 5 english.<BR>\n" );
document.write( "# of ways to select the 4 separaters: 7C4 = 35<BR>\n" );
document.write( "Arrangements of the 5 english: 5!<BR>\n" );
document.write( "Arrangements of the 4 separaters: 4!<BR>\n" );
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document.write( "That leaves 3 books to arrange in 3! ways<BR>\n" );
document.write( "Total arrangements: 5!*35*4!*3* = 604,800<BR>\n" );
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document.write( "So, # no 2 are adjacent = 604800<BR>\n" );
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document.write( "Finally, # at least 2 are adjacent)= 12! - 604800 = 478,396,800<BR>\n" );
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document.write( "Note: Interesting problem.  Let me know when you can <BR>\n" );
document.write( "confirm an answer.<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "This is how far I have gotten:<BR>\n" );
document.write( "(# of ways that all are touching) - (# of ways that none are touching) <BR>\n" );
document.write( "(5!)(7!)(2!) - ???? <BR>\n" );
document.write( "If I am on the right track... how do I figure out the (# of ways that none are touching)?  </SPAN>\n" );
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