[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.