[drenei]

"The results surprised me. Apparently the optimal number of digits to include is 1, and the more digits you have the more likely it is to be registered. In fact, it appears that someone has registered every dot-com combination of 4 digits."

It shouldn't be completely surprising. The more characters (c) than digits (d) in a name, the more options available. 3c1d has more options (26x26x26x10=175760) than 4d (10x10x10x10=10000).

http://en.wikipedia.org/wiki/Factorial

EDIT: And also http://en.wikipedia.org/wiki/Probability_theory

[dododo]

for 3c1d i think a better count is four times higher: you can have aaa1, aa1a, a1aa and 1aaa, which your count of 26x26x26x10 only includes one of these options.

i don't think factorial is relevant here: combinatorially, factorial represents the process of sampling without replacement: so if you were to be able to only use each letter once in each domain, then you would get 26!/4!26! possibilities for 4c.

this is combinatorics: the art of advanced counting, not probability, really.