Thanks Chris, I believe your answer is correct but I'd like to talk about it a different way.
Let the people form a line and the envelopes are on the table.
The first person has a 1 in 5 chance of selecting the right letter. (He selects the correct one)
Now there are 4 on the table so the second person has a 1 in 4 chance of getting the right one (He selects the correct one)
Now there are 3on the table so the second person has a 1 in 3 chance of getting the right one (He selects the correct one)
and so on
so
P(everyone getting the correct letter) = \(\frac{1}{5}\times \frac{1}{4}\times \frac{1}{3}\times \frac{1}{2}\times \frac{1}{1}=\frac{1}{5!}=\frac{1}{120}\)
.