document.write( "Question 1182644: Consider the 720 permutations of the numbers 1, 2, 3, 4, 5, 6. In how many of
\n" ); document.write( "these permutations does 1 appear next to 2 or 3 appear next to 4 or 5 appear
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Algebra.Com's Answer #812746 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Greenestamps' answer of 480 is correct.  Ikleyn has a little trouble with\r\n" );
document.write( "English, as it is not her first language.  If '1 appears NEXT to 2', then\r\n" );
document.write( "and only then, 'numbers 1 and 2 appear next to each other'.\r\n" );
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document.write( "Here is the way I approached it.  I went in the \"back door\".\r\n" );
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document.write( "Let's say each of the numbers have exactly 1 mate.\r\n" );
document.write( "1,2 are a mated pair, 3,4 a mated pair, and 5,6 a mated pair.\r\n" );
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document.write( "We will now enumerate all ways in which no two mates are together and then\r\n" );
document.write( "subtract that result from 720.\r\n" );
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document.write( "We can choose the first number 6 ways.\r\n" );
document.write( "We can then choose the second number 4 ways, as any of its 4 nonmates.\r\n" );
document.write( "That's 24 ways to pick the first two numbers. \r\n" );
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document.write( "In each of those 24 cases, the remaining four consist of a mated pair and a\r\n" );
document.write( "non-mated pair. \r\n" );
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document.write( "So there are two cases for picking the third number.\r\n" );
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document.write( "Case 1.  We pick the third number as the mate of the first number in only\r\n" );
document.write( "one way.\r\n" );
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document.write( "Of the three numbers left, two of them are mates and one is a non-mate to\r\n" );
document.write( "them.  So the non-mated one must go between them.  There are only 2 ways to\r\n" );
document.write( "do that, i.e., to place the mated pair on each side of the non-mated one.\r\n" );
document.write( "That's only 2 ways for case 1. \r\n" );
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document.write( "Case 2.  We pick the third number as a non-mate of the first number (and of\r\n" );
document.write( "course a nonmate to the second number) in 2 ways.\r\n" );
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document.write( "Of the three numbers left, 2 are nonmates to the third number. So we can\r\n" );
document.write( "choose the fourth number in two ways.\r\n" );
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document.write( "Now the last two remaining numbers are neither mates themselves nor are they\r\n" );
document.write( "mates to the fourth number, so they are free to be placed 2 ways, in 5th and\r\n" );
document.write( "6th positions.\r\n" );
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document.write( "That's (2)(2)(2)=8 ways for case 2.\r\n" );
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document.write( "So that's 2+8=10 ways for any of the 24 ways to place the first two numbers.\r\n" );
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document.write( "So the total number of ways no mates are together is (24)(10) or 240 ways.\r\n" );
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document.write( "The means the answer to the problem is 720-240=480 ways.\r\n" );
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document.write( "Edwin
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