||First off, I enjoyed this article and it has given me a few ideas to think about.
I think, though, that it may have tried to get through a bit too much information for its size.
It could very well be that I'm just not that bright - but I found a number of points in the article where, despite being familiar with the concept being discussed, I had to stop and puzzle a while before I could follow the discussion in a meaningful way. I understand that thinking is good, but I wonder if somebody new to the subject would be able to piece things together (especially if they're unfamiliar with the notation).
I understand we don't want every article to start off with a definition of the factorial operator, but I think there needs to be a balance somewhere in terms of the level of the subject being covered and the pacing if the article. I think the article could have benefited from a quick legend for the notation, and more discussion of fewer items.
Oh, also one little bug (Edit: I was confused):
I believe that should be 10^5 ways.
For example, if we have 10 different prizes that need to be divided among 5 people, we can do so in 5^10 ways.
Anywho - interesting article. Thanks.