Sum Rule for Counting

Theorem
If there are $$r_1$$ different objects in the set $$O_1$$, $$r_2$$ different objects in the set $$O_2$$, $$\ldots$$, and $$r_m$$ different objects in the set $$O_m$$, and if $$\cup^{m}_{i=1}O_i=\emptyset$$, then the number of ways to select an object from one of the $$m$$ sets is $$\sum_{i=1}^{m}r_i$$.

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.