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 ```I have 2 decks of standard playing cards (52 cards), shuffled and sitting side by side. I simultaneousely turn 1 card from each deck over and see if they are identical. I repeat this 52 times. What is the probability of, after the 52 turns, there being no match between any of the pairs that have been turned ?```
 ```Hi, milburn-ga: Think of one deck as a permutation (sequential rearrangement) of the other deck, and let's assume that by well shuffling the decks all permutations are equally likely. The property of a permutation that leaves no card in its original position is called being a "derangement": [Derangement] http://www.delphiforfun.org/Programs/Math_Topics/derangements.htm The fraction of permuations that are derangements depends on the number of items being swapped. As the number of items increases, the fraction of permutations which are derangements rapidly approachs 1/e, or roughly 36.788%. Mathematicians often denote the number of derangements of n items by D_n. Recall that the number of permutations of n items (without restrictions) is n!. It can be shown: [Derangements and Applications] http://www.math.uwaterloo.ca/JIS/VOL6/Hassani/hassani5.pdf that D_n is greatest integer less than or equal to: ( n! / e ) + (1/n) for every whole number n > 1. [For the record, D_1 = 0 of course.] Since the fraction you are looking for is D_52 / 52!, it should be clear that the difference between that rational number and the transcendental number 1/e is incredibly small (on the order of 10^-70). If some aspect of my Answer requires Clarification, please ask! regards, mathtalk-ga```