document.write( "Question 1207172: April, Bill , Candace, and Bobby are to be seated at random in a row of
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Algebra.Com's Answer #844899 by AnlytcPhil(1806) $\"\"$ $\"About$

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document.write( "Here is an alternate way to solve the problem using stars and bars.  \r\n" );
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document.write( "Let's suppose the correct interpretation for a successful arrangement is for one\r\n" );
document.write( "of the pair A and B, to be in the far left chair, and the other to be in the far\r\n" );
document.write( "right chair.\r\n" );
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document.write( "There are 2 ways to arrange A and B on the ends.\r\n" );
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document.write( "There are 6*5=30 ways to arrange C and D between them in the 6 chairs between\r\n" );
document.write( "them.\r\n" );
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document.write( "So the number of SUCCESSFUL arrangements is 2*6*5=60.\r\n" );
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document.write( "Now let's look at the number of POSSIBLE arrangements. Take this sample\r\n" );
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document.write( "CDBA.  There are 4!=24 arrangements like this one.\r\n" );
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document.write( "We must now insert 4 empty chairs among them. \r\n" );
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document.write( "This involves the number of partitions of 4 of length 5.\r\n" );
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document.write( "We make a row of 4 stars and 4 bars, like this sample:\r\n" );
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document.write( "**||*|*|\r\n" );
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document.write( "That is the partition 2+0+1+1+0=4.  That's because \r\n" );
document.write( "there are: \r\n" );
document.write( "2 stars left of the 1st bar, \r\n" );
document.write( "0 stars between the 1st and 2nd bar,\r\n" );
document.write( "1 star between the 2nd and 3rd bar,\r\n" );
document.write( "1 star between the 3rd and 4th bar, and\r\n" );
document.write( "0 stars after the 4th bar.\r\n" );
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document.write( "This sample partition of 4 of length 5, which is 2+0+1+1+0 makes the sample \r\n" );
document.write( "ADCB to become:\r\n" );
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document.write( "_ _ A D _ C _ B\r\n" );
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document.write( "where the blanks represent the 4 empty chairs.\r\n" );
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document.write( "Now we find the number of ways to insert the 4 empty chairs among the 4 people.\r\n" );
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document.write( "There are 4 indistinguishable stars and 4 indistinguishable bars.\r\n" );
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document.write( "So there are  ways to put 4 empty chairs \r\n" );
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document.write( "among the 4 people.\r\n" );
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document.write( "So let's put what we've said here all together: \r\n" );
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document.write( "There are 4!=24 ways to form something like this: ADCB.\r\n" );
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document.write( "Then there are 70 ways to form something like this _ _ A D _ C _ B\r\n" );
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document.write( "That's 24x70 = 1680 possible ways.\r\n" );
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document.write( "So the probability is \r\n" );
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document.write( "Edwin

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