Question

How many terms are needed when the inclusion exclusion principle is used to express the number of elements in the union of seven sets if no more than five of these sets have a common element?

Short answer

The total terms are needed when Inclusion-Exclusion principal is used to express the number of elements in the union of seven sets if no more than five of these sets have a common element is\(119\).

Step-by-step solution

In the problem given

Union of seven sets.

The definition and the formula for the given problem

Principle of Inclusion and Exclusion is an approach which derives the tactic of finding the number of elements within the union of two finite sets. this is often used for solving combinations and probability problems when it's necessary to seek out a counting method, which makes sure that an object isn't counted twice.

Determining the sum in expanded form

For the union of \(n\) sets, the total term required by the Inclusion-Exclusion principal is \(\sum\limits_{j = 1}^n {\left( {\begin{array}{*{20}{l}}n\\j\end{array}} \right)} \) terms.

For the union of seven sets, as no more than five of these sets have common elements, which implies \(\sum\limits_{i = 1}^5 {\left( {\begin{array}{*{20}{l}}7\\i\end{array}} \right)} = {2^7} - 1 - 7 - 1 = 119\) terms.

Hence, the total terms are needed when Inclusion-Exclusion principal is used to express the number of elements in the union of seven sets if no more than five of these sets have a common element is\(119\).