For an indeterminate $q$ and integers $n \ge m \ge 0$ we define the following:

(a) $(m)_q = q^{m-1} + q^{m-2} + \cdots + 1$ for $m>0$

(b) $(m!)_q = (m)_q (m-1)_q \cdots (1)_q$ for $m>0$ and $(0!)_q = 1$

(c) ${n \choose m}_q = \frac{(n!)_q}{(m!)_q ((n-m)!)_q}$ If $m>n$ then we define ${n \choose m}_q=0$

The expressions ${n \choose m}_q$ are called $q$ binomial coefficients or Gaussian polynomials.

Note: if we replace $q$ with 1, then we obtain the familiar integers, factorials, and binomial coefficients. Specifically,

(a) $(m)_1 = m$

(b) $(m!)_1 = m!$

(c) ${n \choose m}_1 = {n \choose m}$

(d) ${m \choose m}_q=1$