Rule of sum

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In combinatorics, the rule of sum is a basic counting principle. Stated simply, it is the idea that if we have a ways of doing something and b ways of doing another thing and we can not do both at the same time, then there are a + b ways to choose one of the actions.

More formally, the rule of sum is a fact about set theory. It states that sum of the sizes of a finite collection of pairwise disjoint sets is the size of the union of these sets. That is, if $S 1, S 2,..., S n$ are pairwise disjoint sets, then we have:

$|S_{1}|+|S_{2}|+\cdots+|S_{n}| = |S_{1} \cup S_{2} \cup \cdots \cup S_{n}|$