Derivation of Derangement formula

This question comes from my unsuccessful attempt to solve

http://www.mathhelpforum.com/math-he...-problems.html

(Reason im posting a new thread: dont want to hijack the other one with my own questions).

The question was:

"there are n letters and n addressed envelopes. Letters are placed randomly in envelopes. find the probability that at least 1 letter goes in the correct envelope"

The correct answer is

Quote:

Originally Posted by

**Plato** The number of derangements on

items is

.

The probability that at least one term of a permutation remained fixed is

.

But i thought it was

Quote:

Originally Posted by

**SpringFan25** you can do this with a tree diagram, where each branch has 2 possible outcome (correct, not correct).

I wouldn't try and draw the whole thing, but its good to ahve in the back of your head.

**Step 1:**
Find the probability that no paycheck is in the correct envelope:

P(0) =

**Step 2:**
P(at least 1) = 1-P(0)

Check for n=3:

, which you said was correct

So, i have 2 questions:

1) What's the conceptual mistake in my approach

2) Is there a derivation of the subfactorial function somewhere?