Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Discrete mathematics / 1 / a / iii / Solution

The rule of product says that a sequence of $k$ choices, with $n_i$ possibilities at stage $i$ regardless of earlier choices, has ${\prod }_{i=1}^k{n}_i$ possible outcomes. Applied to the Cartesian product of finite sets, it gives $|A\times B|=|A||B|$. Here independence means that the available choices do not restrict one another; no probability assumption is needed.