  Gaussian binomial coefficients are q-analogues of the binomial coefficients
of integers. On the other hand, binomial coefficients have been extended to
finite words, i.e., elements of the finitely generated free monoids. In this
paper we bring together these two notions by introducing q-analogues of
binomial coefficients of words. We study their basic properties, e.g., by
extending classical formulas such as the q-Vandermonde and Manvel's et al.
identities to our setting. As a consequence, we get information about the
structure of the considered words: these q-deformations of binomial
coefficients of words contain much richer information than the original
coefficients. From an algebraic perspective, we introduce a q-shuffle and a
family q-infiltration products for non-commutative formal power series.
Finally, we apply our results to generalize a theorem of Eilenberg
characterizing so-called p-group languages. We show that a language is of this
type if and only if it is a Boolean combination of specific languages defined
through q-binomial coefficients seen as polynomials over $\mathbb{F}_p$.
