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 $\displaystyle \bigcap_{i=1}^m S_i = \varnothing$.

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

Proof

A direct application of Cardinality of Set Union.

$\blacksquare$

Sources

• A. Tucker: Applied Combinatorics 5th Edition (2007)