In article <email@example.com>, <firstname.lastname@example.org> wrote:
> Thought experiment: > > Lets say that I randomly draw N letters out of a Scrabble letter-bag. I will then > place those letters in a horizontal line on a table. How many UNIQUE patterns of > letters can I produce? If there were no duplicates it would be easy (N factorial) > but when there are duplicate letters it seems more complicated.
Let's start with two letters, you have "m" of one and "n" of the other. So the total number to place them is (m+n)!, but you have to divide by m! to account for the fact that all different orderings of the "m's" yield the same word, and you have to also divide by n! for the same reason; so it's
(m+n)! ------ m!n!
You can also think about it as choosing "m" place for the first letters and "n" for the other, so it's clearly (n+m choose n).