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 $\ds \bigcap_{i \mathop = 1}^m S_i = \O$.

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

Proof
A direct application of Cardinality of Set Union.