document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183400>Question 1183400</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813700 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=813700\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> UPDATE #2!\r<BR>\n" );
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document.write( "To tutor @ikleyn:\r<BR>\n" );
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document.write( "You totally didn't understand the strategy I laid out.  The scenario I envisioned was, fixing all left socks on one row, such as in the case of 3 pairs, <BR>\n" );
document.write( "one will have the initial row\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B1%5D ALIGN=MIDDLE  ALT=\"L%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >          <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B2%5D ALIGN=MIDDLE  ALT=\"L%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >              <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B3%5D ALIGN=MIDDLE  ALT=\"L%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and then inserting R1, R2, and R3, one after the other, in ANY of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight+%28FOUR%29 ALIGN=MIDDLE  ALT=\"highlight+%28FOUR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > slots available for insertion, and nowhere else! <BR>\n" );
document.write( "This is for the purpose of producing arrangements such that an R sock will not be either to the left or right of its L match.<BR>\n" );
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document.write( "The arrangement you laid out is not my interpretation, it is YOURS, and yours alone, <BR>\n" );
document.write( "hence you have found contradiction only in your approach, but not mine.  SO you destroyed your own logical construction.\r<BR>\n" );
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document.write( "UPDATE! \r<BR>\n" );
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document.write( "Tutor @ikleyn claims that the problem is NOT about derangement.  But if you looked at and even understood what is presented at <BR>\n" );
document.write( "the wikipedia page she cited (just like I did), then you will realize that IT IS a problem on derangement.  <BR>\n" );
document.write( "The strategy is to lay out one from each pair of the 12 pairs of socks on a row and then to insert each of the other 12 socks next to the others laid out.  <BR>\n" );
document.write( "We have to count the number of ways that an insertion doesn't produce a \"match\", i.e., a sock doesn't lie beside its match.  <BR>\n" );
document.write( "It then becomes a problem similar to 12 men each of which not getting his correct hat back from the hatstand containing 12 hats. <BR>\n" );
document.write( "Of course, one may argue that the sock might be inserted either to the left or to the right of a fixed sock, but this only doubles the permutations overall. <BR>\n" );
document.write( "Incidentally, this will also double the number of drangements that are possible, so the factor 2 just cancels out.\r<BR>\n" );
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document.write( "And to tutor @ikleyn, if you wanted to help a student sincerely, don't just say another tutor is wrong -- give the correct answer and solution! <BR>\n" );
document.write( "Otherwise all your effort is useless and has no value pedagogically.\r<BR>\n" );
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document.write( "I think it is safe to say that tutor @ikleyn DOES NOT KNOW the answer, nor has any idea on how to solve the problem.\r<BR>\n" );
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document.write( "This is a problem on the derangements of n = 12 objects, with no sock lying next to its match.\r<BR>\n" );
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document.write( "The answer is (!12)/12! = 176,214,841/12! = 0.367879, to 6 d.p. This is almost equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fe ALIGN=MIDDLE  ALT=\"1%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "***!n is called the number of drangements of n objects, where none of the n objects are paired correctly with its right match.  <BR>\n" );
document.write( "You are referred to https://en.wikipedia.org/wiki/Derangement. </SPAN>\n" );
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