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There is a nice way that I learnt learned from Martin Gardner's books when I was young(er).

Imagine the following situation. There is a party with $N$ people. All of them throw their hats in the middle of the room, and then each of them takes one hat randomly. What is the probability that nobody gets its own hat?

The surprising answer is that this probability equals

$1-1+\frac{1}{2!}-\frac{1}{3!} + \cdots + \frac{(-1)^N}{N!}$,

which goes to $\frac{1}{e}=0.36788...$ for $N \to \infty$.

1

There is a nice way that I learnt from Martin Gardner's books when I was young(er).

Imagine the following situation. There is a party with $N$ people. All of them throw their hats in the middle of the room, and then each of them takes one hat randomly. What is the probability that nobody gets its own hat?

The surprising answer is that this probability equals

$1-1+\frac{1}{2!}-\frac{1}{3!} + \cdots + \frac{(-1)^N}{N!}$,

which goes to $\frac{1}{e}=0.36788...$ for $N \to \infty$.