  In this article, we offer group-theoretic, field-theoretic, and topological
interpretations of the Gaussian binomial coefficients and their sum. For a
finite $p$-group $G$ of rank $n$, we show that the Gaussian binomial
coefficient $\binom{n}{k}_p$ is the number of subgroups of $G$ that are
minimally expressible as an intersection of $n - k$ maximal subgroups of $G$,
and their sum is precisely the number of subgroups that are either $G$ or an
intersection of maximal subgroups of $G$. We provide a field-theoretic
interpretation of these quantities through the lens of Galois theory and a
topological interpretation involving covering spaces
