We have a standard 52-card deck. The cards positions are listed, then this 52-card deck was well-shuffled. What is the probability that there is no card remains at its initial position?
The probability that there is no card remains at its initial position when a standard 52-card deck is well shuffled
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Best Answer
The probability in question is $$ \frac {!52}{52!}=\sum_{k=0}^{52}\frac {(-1)^k}{k!}\approx\frac1e, $$ where $!n $ is the number of permutations of $n $ objects without fixed points also known as subfactorial of $n $.