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 $S_1,\ldots,S_n$ are pairwise disjoint sets, then we have:

$|S_1|+\cdots+|S_n|=|S_1\cup\cdots\cup S_n|$