Sum Rule for Counting
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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.
$\blacksquare$