Sum Rule for Counting

Theorem
Let there be:
 * $r_1$ different objects in the set $S_1$
 * $r_2$ different objects in the set $S_2$
 * $\ldots$
 * $r_m$ different objects in the set $S_m$.

Let $\displaystyle \bigcup_{i=1}^m S_i = \varnothing$.

Then the number of ways to select an object from one of the $m$ sets is $\displaystyle \sum_{i=1}^m r_i$.

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

The sum $a+b$ is the number of items resulting when a set of $a$ items is added to a set of $b$ items.