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 $$i=1,\ldots ,m$$).

If the number of outcomes at each stage is independent of the choices in previous stages and if the composite outcomes are 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 $$ab$$ (for $$a,b\in \mathbb{N}^*$$) is the number of sequences $$(A,B)$$, where $$A$$ can be any one of $$a$$ items and $$B$$ can be any one of $$b$$ items.