Approximation to Derangement is Closest Integer

 

We now show that for ,  is the closest integer to ; it is straightforward to use a spreadsheet to verify this result for many cases.   We look at the absolute value of the difference between  and :

 and with a little rearrangement, we have

thus , and for ,  is the closest integer to .

 

 

 

Note: This is a supplement to, Borkovitz, Debra, “The Name Game: Exploring Random Permutations.”  Mathematics Teacher 98 (October 2005).

 

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