A secretary addresses 50 different letters and envelopes to 50 different people, but the letters are randomly mixed before being put into envelopes. What is the probability that at least one letter gets into the correct envelope?

This is known as a derangement.

The probability that no letter ends up in its envelope is

\sum_{k=0}^{N}\frac{(-1)^{k}}{k!}

This is the series for 1/e. So, for large N, the probability approaches around 37%.

So, for the probability that at least one gets in its envelope would be 1 minus the above.

1-\sum_{k=0}^{50}\frac{(-1)^{k}}{k!}\approx .632

About 63.2%

Cool! I'd never heard of derangement before (at least not this particular definition - lol!).