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Rule of Sum and the Inclusion-Exclusion Principle

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If there are ways of getting a result and ways to get a result , then the number of ways of getting or is , as long as the results and do not overlap.

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To get the right number when there is overlap, think of the possible results and as sets. Then the number of ways to get an element from or is . This is called the principle of inclusion-exclusion.

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Contributed by: George Beck (July 2018)
Open content licensed under CC BY-NC-SA

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The sum rule generalizes when there are more than two kinds of results, giving

,

as long as all the pairwise intersections

, , ,

are empty.

If there are overlaps, the right-hand side of the formula is an alternating sum. For example, here is the formula for three sets:

.

Reference

[1] Wikipedia. "Rule of Sum." (Jun 4, 2018) en.wikipedia.org/wiki/Rule_of_sum.

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