Product Rule for Counting

Theorem
Suppose a process can be broken into $$m$$ successive, ordered, stages, with the $$i^{\text{th}}$$ stage having $$r_i$$ possible outcomes (for $$i = 1, \ldots, m$$).

Let the number of outcomes at each stage be independent of the choices in previous stages

Let the composite outcomes be all distinct.

Then the total procedure has $$\prod_{i=1}^m r_i$$ different composite outcomes.

Proof
The validity of this rule follows directly from the definition of multiplication of integers.

The product $$a b$$ (for $$a, b \in \N^*$$) is the number of sequences $$\left({A, B}\right)$$, where $$A$$ can be any one of $$a$$ items and $$B$$ can be any one of $$b$$ items.