Product Rule for Counting

Theorem
Let it be possible to choose an element $\alpha$ from a given set $S$ in $m$ different ways.

Let it be possible to choose an element $\beta$ from a given set $T$ in $n$ different ways.

Then the ordered pair $\tuple {\alpha, \beta}$ can be chosen from the cartesian product $S \times T$ in $m n$ different ways.

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

The product $a b$ (for $a, b \in \N_{>0}$) is the number of sequences $\sequence {A, B}$, where $A$ can be any one of $a$ items and $B$ can be any one of $b$ items.

Also known as
Some sources give this as the General Combinatorial Principle.

Some sources call it the $r$, $s$ principle: if one operation can be performed in $r$ different ways, and if another operation can be performed in $s$ different ways, the two operations can be performed in succession in $r \times s$ different ways.

Also see

 * Fundamental Principle of Counting