This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an $F\_1$ case of $q$-binomial coefficients and identities? Here $F\_1$ is the field of one element.



Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients?
Yes, it is not only fair, but also often very useful. The analogy is that sets are like vector spaces over $\mathbb F\_1$, so for example the analog of $\binom{n}{k}$ counting $k$-element subsets of an $n$-element set, is that $\binom{n}{k}\_q$ counts the number of $k$-dimensional subspaces of $\mathbb F\_q^n$.


In lack of a rigorous theory of $\mathbb F\_1$, this usefulness can be illustrated through numerous examples. Take for example the [Erdos-Ko-Rado theorem](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem), which is an important starting point of extremal combinatorics. Even though writing down the analogous statement over $\mathbb F\_q$ is very easy, by just interchanging subsets with subspaces, the proof turned out to be much harder (the first proof was due to Frankl and Wilson).


The only uniform proofs I know Erdos-Ko-Rado over all characteristics (including characteristic 1) are "geometric" in nature and were discovered much later. However I'm afraid this isn't particularly emphasized in the literature. For example I haven't seen anywhere explicitly mention the analogy between the method of "shadows" and the study of the relevant partial flag variety over $\mathbb F\_q$. Still, I think this is a very fun example of interplay between combinatorics of sets and geometry over $\mathbb F\_q$ that has produced non-trivial theorems.


