Given some set of unique elements of length n. What is the probability that if you randomly shuffle these elements, none of them will be in their original position?
2026-04-25 03:13:33.1777086813
Probability of Completely New Ordering
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A permutation in which no element appears in its original position is called a "derangement". The topic of derangements has been thoroughly studied, and you can read about it on Wikipedia. In particular, the probability that a permutation of $n$ elements is a derangement is:
$$\sum_{i=0}^n \frac{(-1)^i}{i!}.$$