# 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.

$\blacksquare$