document.write( "Question 1183400: Suppose you have n different pairs of socks (n left socks and n right socks, for 2n individual socks total) in your dresser. You take the socks out of the dresser one by one without looking and lay them out in a row on the floor. What is the probability that no two matching socks are next to each other? \n" ); document.write( "

Algebra.Com's Answer #813708 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            It is DEFINITELY NOT derangement.\r
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\n" ); document.write( "\n" ); document.write( "            Tutor @robertb incorrectly cites the definition of the derangement permutations,\r
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\n" ); document.write( "\n" ); document.write( "See the referred Wikipedia article on derangement permutations\r
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\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Derangement\r
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\n" ); document.write( "\n" ); document.write( "            My text below is written in response to notes by @robertb.\r
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\n" ); document.write( "\n" ); document.write( "OK, it looks like I should explain, why I did not solve it and why I think it is not derangement problem.\r
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document.write( "Let assume that we have two pairs of socks, N1 and N2.  Only two pairs, for simplicity.\r\n" );
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document.write( "Let the pair N1 is the pair  (1L,1R) : N1 left sock  and  N1 right sock.\r\n" );
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document.write( "Let the pair N2 is the pair  (2L,2R) : N2 left sock  and  N2 right sock.\r\n" );
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document.write( "We start from this arrangement\r\n" );
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document.write( "    1L,1R,    2L,2R.        (1)\r\n" );
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document.write( "All your interpretation is based on considering the pairs like\r\n" );
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document.write( "   (1L,1R),  (2L,2R)        (2)\r\n" );
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document.write( "and their derivatives.\r\n" );
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document.write( "To make derangement, I transpose sock 2R from the second pair to the first pair \r\n" );
document.write( "and transpose sock 1R from the first pair to the second.  I get\r\n" );
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document.write( "   (1L,2R),  (2L,1R).       (3)\r\n" );
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document.write( "In your interpretation, two pairs in (3) are deranged: you count it as deranged.\r\n" );
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document.write( "Let's write (3) as a row of socks  (without separating them in pairs)\r\n" );
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document.write( "    1L, 2R, 2L, 1R.\r\n" );
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document.write( "But now two socks 2R and 2L do belong to one pair and are next to each other, so this arrangement \r\n" );
document.write( "can not be counted as deranged, and the logical construction is destroyed.\r\n" );
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document.write( "So, your interpretation has an interior contradiction.\r\n" );
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document.write( "It is WHY I could not solve the problem and it is why I think, that the problem, \r\n" );
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document.write( "as it is worded, printed, posted and presented, is DEFECTIVE and can not be solved in terms of derangement.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Long time after my previous posts (several months after), I found the solution for closely related TWIN problem in the Internet.\r
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\n" ); document.write( "Bridget works part-time in a shoe store. Sometimes when it is not busy, she rearranges the shoes for fun.
\n" ); document.write( "If she takes six different pairs of shoes and rearranges them in a row, in how many ways can she rearrange them
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\n" ); document.write( "\n" ); document.write( "Find the solution at this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.quora.com/In-how-many-ways-can-you-rearrange-6-different-pairs-of-shoes-in-a-row-so-that-no-two-shoes-match\r
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\n" ); document.write( "\n" ); document.write( "ANSWER. The number of ways is 168,422,400 (for n= 6 pairs).\r
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