document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206447>Question 1206447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Six story books, A, B, C, D, E, and F were arranged on a shelf randomly. If A and B were not placed together, what is the probability that there was one book between A and B? </SPAN>\n" );
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document.write( "The number of ways the 6 books can be arranged on the shelf is 6! = 720.<br><BR>\n" );
document.write( "For the number of ways the two books A and B can be together, consider them as a single unit.  Then there are 5 things to be arranged on the shelf, in 5! = 120 different ways; and within the AB pair those two can be arranged in 2 different ways.  So the number of ways the books can be arranged with A and B together is 120*2 = 240.<br><BR>\n" );
document.write( "So the number of ways in which the books can be arranged with A and B NOT together is 720-240 = 480.<br><BR>\n" );
document.write( "For the case where there is exactly one book between A and B....<br><BR>\n" );
document.write( "(1) There are 4 places in the arrangement of the 6 books where there is exactly one book between A and B -- positions 1 to 3, 2 to 4, 3 to 5, and 4 to 6.<BR>\n" );
document.write( "(2) The two books A and B can be in either of 2 different orders.<BR>\n" );
document.write( "(3) The other four books can be arranged in 4! = 24 different ways.<br><BR>\n" );
document.write( "The number of ways for there to be one book between A and B is 4*2*24 = 192.<br><BR>\n" );
document.write( "So the probability that there is one book between A and B, given that A and B are not together, is 192/480 = 2/5.<br><BR>\n" );
document.write( "ANSWER: 2/5<br><BR>\n" );
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