(In order that this question appears as answered I'm converting Richard's comment into an answer.)

The (somewhat nonstandard) convention in the text, stated just before the corollary, is that $[n]!_q = (1-q)(1-q^2)\cdots(1-q^n)$. For the quantity $[n]!_q/(1-q)^n$, the notation $\mathbf{(n)!}$ is used instead.