Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients? 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.
 A: 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, 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.
