document.write( "Question 1206447: 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? \n" ); document.write( "

Algebra.Com's Answer #843937 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Six story books, A, B, C, D, E, and F were arranged on a shelf randomly.
\n" ); document.write( "If A and B were not placed together, what is the probability that there was one book between A and B?
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document.write( "(1)  In all, there are 6! = 6*5*4*3*2*1 = 720 different permutations of 6 books on the shelf.\r\n" );
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document.write( "(2)  Of them, there are 2*5! = 240 permutations, where the books A and B are together.\r\n" );
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document.write( "(3)  Hence, in the rest 720 - 240 = 480 permutations, the books A and B are not together.\r\n" );
document.write( "     It is the number of all possible permutations under the condition \"If A and B were not placed together\".\r\n" );
document.write( "     So, 480 is the denominator in the future fraction for the probability.\r\n" );
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document.write( "(4)  The number of all possible permutations of 6 letters with the block \r\n" );
document.write( "     of 3 letters ACB in 3 positions between #1 and #6 is 4! = 24.\r\n" );
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document.write( "(5)  The number of all possible permutations of 6 letters with the block \r\n" );
document.write( "     of 3 letters AXB in 3 positions between #1 and #6 with X= C or D or E or F \r\n" );
document.write( "     is 4*4! = 4*24 = 96.\r\n" );
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document.write( "(6)  The number of all possible permutations of 6 letters with the block \r\n" );
document.write( "     of 3 letters BXA in 3 positions between #1 and #6 with X= C or D or E or F \r\n" );
document.write( "     is also 4*4! = 4*24 = 96.\r\n" );
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document.write( "(7)  So, the numerator in the fraction for probability is  96+96 = 192.\r\n" );
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document.write( "(8)  Finally, the probability under the problem's question is  \r\n" );
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document.write( "         =  =  =  = 0.4.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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