# What is the hat number problem?

I believe I remember the answer is surpringly $\displaystyle \frac{1}{e}$ when calculating some permutation when people are switching hats. Do you know what I'm talking about?

It's supposedly applied mathematics where a number of people are switching hats and surprisingly a probability turn out to be $\displaystyle \frac{1}{e}$ or likewise.

## Update

Well I found it and I think it's surpring that it's 1/e: http://books.google.se/books?id=OVkoCcszEZ0C&pg=PA39&dq=hat&redir_esc=y#v=onepage&q=hat&f=false

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2
Maybe look at the Wikipedia article on derangements. – André Nicolas Jan 30 at 7:34

1
– Glen_b Jan 30 at 7:48

## 1 Answer

At a party $n$ men take off their hats. The hats are then mixed up and each man randomly selects one. What is the probability that no man select their own hat. Also show that if $n$ is tends to $\infty$, then the probability will become $1 \over e$.
To solve this problem use Poincare's Theorem. Normally your probability will be $$p=\sum_{k=0}^n\frac{(-1)^k}{k!}$$ and note that $$\lim_{n \to \infty} \sum_{k=0}^n \frac{(-1)^k}{k!}= \frac{1}{e}$$
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