**Inclusion/exclusion principle
**

The inclusion/exclusion principle is a counting technique that relates information about the number of elements in a set, its subsets, and the intersections of those subsets:

In a set with some subsets that intersect in some way (possibly not at all), the number of elements that are *not* in any of the subsets will equal:

the number of elements in the original set

minus the sum of the numbers of elements in all of the subsets,

plus the sum of the numbers of elements in the intersections of all pairs of the subsets

minus the sum of the numbers of elements in the intersections of all triples of the subsets

plus the sum of the numbers of elements in the intersections of all quadruples of the subsets

plus. . . minus. . .

Here is an example:

The Venn diagram below illustrates a set *S,* which contains 90 elements and 5 subsets, *A, B, C, D,* and *E.*

A | 33 | intersection of A, C, D | 3 |

B | 35 | Intersection of A, D, E | 0 |

C | 30 | intersection of A, B, D | 8 |

D | 46 | intersection of A, B, E | 0 |

E | 10 | intersection of A, C, E | 0 |

A intersection B | 18 | intersection of A, D, E | 0 |

A intersection C | 10 | intersection of B, C, D | 6 |

A intersection D | 9 | intersection of B, D, E | 0 |

A intersection E | 0 | intersection of B, C, E | 0 |

B intersection C | 9 | intersection of C, D, E | 0 |

B intersection D | 21 | intersection of A, B, C, D | 2 |

B intersection E | 0 | intersection of A, B, C, E | 0 |

C intersection D | 17 | intersection of A, B, D, E | 0 |

C intersection E | 3 | intersection of A, C, D, E | 0 |

D intersection E | 5 | intersection of B, C, D, E | 0 |

intersection of A, B, C | 5 | intersection of A, B, C, D, E | 0 |

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