Basic Probability Question

Hi everyone.

I last did probability about 5yrs ago(?) and I've been trying to refresh my memory of it. Here's a question that has me stumped.

How many ways are there of placing n un-identifiable letters in m mailboxes?

If you can distinguish the letters the answer is quite easy: m^n, because each letter has m possible boxes it can go into.
For this case where the letters are un-identifiable or un-distinguishable, the only way I can think of doing it is the following:

If you have 1 letter there are m ways.

If you have 2 letters there are m ways of putting them all in one box, plus m!/[2!(m-2)!] ways of putting one letter in its own box (m is clearly >= 2 otherwise the problem is trivial).

If you have 3 letters there are m ways of putting all the letters in one box, plus m(m-1) ways of putting 2 letters in one box and the other elsewhere, plus m!/[3!(m-3!)] ways of putting each letter in its own box (provided m >= 3).
Etc. on to higher numbers of letters.

Is this correct? Is there a simpler way of going about it? I ask if there is a simpler way because the case where we can distinguish the letters appears so much easier.

Thank you for any help.