Question 1183400
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            It is  DEFINITELY  NOT  derangement.



            Tutor @robertb incorrectly cites the definition of the derangement permutations,

            and incorrectly tries to use this notion/conception.



See the referred Wikipedia article on derangement permutations


https://en.wikipedia.org/wiki/Derangement



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            My text below is written in response to notes by @robertb.

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OK,  it looks like I should explain,  why I did not solve it and why I think it is not derangement problem.



<pre>
Let assume that we have two pairs of socks, N1 and N2.  Only two pairs, for simplicity.


Let the pair N1 is the pair  (1L,1R) : N1 left sock  and  N1 right sock.

Let the pair N2 is the pair  (2L,2R) : N2 left sock  and  N2 right sock.


We start from this arrangement

    1L,1R,    2L,2R.        (1)


All your interpretation is based on considering the pairs like

   (1L,1R),  (2L,2R)        (2)

and their derivatives.


To make derangement, I transpose sock 2R from the second pair to the first pair 
and transpose sock 1R from the first pair to the second.  I get

   (1L,2R),  (2L,1R).       (3)


In your interpretation, two pairs in (3) are deranged: you count it as deranged.


Let's write (3) as a row of socks  (without separating them in pairs)

    1L, 2R, 2L, 1R.


But now two socks 2R and 2L do belong to one pair and are next to each other, so this arrangement 
can not be counted as deranged, and <U>the logical construction is destroyed</U>.


So, your interpretation has an interior contradiction.


It is WHY I could not solve the problem and it is why I think, that the problem, 

as it is worded, printed, posted and presented, is DEFECTIVE and can not be solved in terms of derangement.
</pre>


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Long time after my previous posts &nbsp;(several months after), &nbsp;I found the solution for closely related &nbsp;TWIN &nbsp;problem in the &nbsp;Internet.


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Bridget works part-time in a shoe store. Sometimes when it is not busy, she rearranges the shoes for fun. 
If she takes six different pairs of shoes and rearranges them in a row, in how many ways can she rearrange them 
so that no two shoes match?
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Find the solution at this link


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



The referred text contains hidden parts. When you will read it, open these hidden parts of the text.


<U>ANSWER</U>.   &nbsp;&nbsp;The number of ways is  &nbsp;168,422,400  &nbsp;&nbsp;(for &nbsp;n= &nbsp;6 &nbsp;pairs).