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 $\displaystyle \bigcap_{i \mathop = 1}^m S_i = \varnothing$.
Then the number of ways to select an object from one of the $m$ sets is $\displaystyle \sum_{i \mathop = 1}^m r_i$.
Proof
A direct application of Cardinality of Set Union.
$\blacksquare$