Why does the expansion of $e^x$ appear to arise in the formula for derangement of $n$ things $D_{n}=n!\sum_{k=0}^n \frac{(-1)^k}{k!}$

The following is very heuristic:

The probability that a given one of the $n$ things is not at his proper place is $1-{1\over n}$. Assuming that the $n$ things are placed properly or improperly independently of each other the probability that all things are improperly placed computes to $$\left(1-{1\over n}\right)^n\doteq{1\over e}\ .$$

Tags:

Combinatorics

Taylor Expansion

Derangements

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