Random Allocation of $ N $ Objects to $ N$ sites

An example of $e$ in a random allocation problem
The problem is easily stated.
Consider a set of $N$ allocations $\{A_1, \dots A_i , \dots A_N\}$ and a set of $N$ objects $\{o_1, \dots o_i , \dots o_N\}$. Then allocate each object at random and independently with probability $\frac{1}{N}$ to the set of allocations. The in the limit that $N \rightarrow \infty$ what is proportion of empty allocations?

For the finite $N$ allocations problem the probability of any one allocation the probability $P_N$ of not being occupied after the $N$ objects are allocated is

$$P_N = (1- \frac{1}{N})^N$$

and this is just the proportion of unoccupied allocations in the limit $N \rightarrow \infty$. Therefore using

$$P_N = \exp ( \log (P_N))$$

$$P_N = \exp N( \log ( 1- \frac{1}{N}))$$

$$P_N = \exp N((- \frac{1}{N}) + O(\frac{1}{N}^2))$$

$$ \lim_{N \rightarrow \infty} P_N = \frac{1}{e}$$.

Someone must have noticed this before but I have not come across it elsewhere.

The context in which this curiosity was found was looking at some problem of surface contamination in ultra high vacuum systems.

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