Principle of Inclusion and Exclusion


The Principle of Inclusion and Exclusion (PIE) is useful for counting the number of elements in the union of multiple sets. In particular, it is useful for the union of sets that are not mutually exclusive, where adding the number of elements in each set does not give the correct value.

For example, consider the sets \(A = \{1,2,3\}\) and \(B = \{3, 4, 5\}\). In this case, adding the number of elements in each set gives \(6\), when in reality \(\lvert A \cup B \rvert = 4\).

Below, cases of PIE for \(2\), \(3\), and \(n\) sets will be described.

PIE for 2 sets

For the case with \(2\) sets, PIE is often called the Addition Rule, and can be stated as:

\[|A \cup B| = |A| + |B| - |A \cap B|\]

The subtraction on the right side of the equation is necessary to prevent double-counting of \(A \cap B\), which can be seen visually in the diagram below:

Two sets that are not mutually exclusive

PIE for 3 Sets

For \(3\) sets, PIE can be stated as:

\[|P \cup M \cup F| = |P| + |M| + |F| - |P \cap M| - |P \cap F| - |M \cap F| + |P \cap M \cap F|\]

In this equation, the first \(3\) subtractions eliminate repeated countings of the intersection of every pair of sets. However, in the process, the intersection of all \(3\) sets is wrongly ignored, which is why \(|P \cap M \cap F|\) must be added.

Three sets that are not mutually exclusive

PIE for n Sets

Generalizing from PIE for \(2\) and \(3\) sets, PIE for \(n\) sets can be written as:

\[\left\lvert \bigcup_{i=1}^{n} A_{n} \right\rvert ={} \sum_{i=1}^{n} \lvert A_{i} \rvert \, - \sum_{1 \leq i \leq j \leq n} \lvert A_{i} \cap A_{j} \rvert \, + \sum_{1 \leq i \leq j \leq k \leq n} \lvert A_{i} \cap A_{j} \cap A_{k} \rvert \, - \ldots \, + (-1)^{n-1} \lvert A_{1} \cap A_{2} \cap \ldots \cap A_{n} \rvert\]

Example Problem 1

How many permutations of \(26\) letters (a-z) do not contain any of: fish, rat, bird?

Let \(F\), \(R\), and \(B\) be sets of permutations that contain fish, rat, and bird, respectively. We begin by computing the cardinality of these sets, as well as their intersections:

\[\begin{split} |F| &= (26 - 4 + 1)! = 23! \\ |R| &= (26 - 3 + 1)! = 24! \\ |B| &= (26 - 4 + 1)! = 23! \\ |F \cap R| &= (26 - 4 - 3 + 1 + 1) = 21! \\ |F \cap B| &= 0 \\ |R \cap B| &= 0 \\ |F \cap R \cap B| &= 0 \\ \end{split}\]

The last \(3\) cardinalities are \(0\) because the words contain the same letters and cannot appear in the same permutation. Then, we apply PIE to calculate the number of permutations containing fish, rat, or bird:

\[\begin{split} |F \cup R \cup B| &= |F| + |R| + |B| - |F \cap R| - |F \cap B| - |R \cap B| + |F \cap R \cap B| \\ &= 23! + 24! + 23! - 21! \end{split}\]

Finally, we subtract this value from the total number of permutations, giving \(26! - \left[ (2)23! + 24! - 21! \right]\) as the final answer.

Example Problem 2

How many ways can you seat n couples at a very long dinner table with seats 1..2n such that no couple sits next to each other.