22.1: Motivation

Example \(\PageIndex{1}\)

How many different four-member study groups could be formed from a class of twenty students?

Solution

We will use the Division Rule, first imposing additional structure on each possible study group. Within a study group, create positions of President, Vice-President, Secretary, and Janitor (cards, anyone?). Then there are \(P(20, 4)\) such structured groups.

But a real study group doesn't have this structure, so we'll consider two structured groups to be equivalent when they have the same membership, regardless of positions. How many equivalent structured groups with a given membership are there? Within a group of four, the additional structure is just an ordering, and the number of orderings of a given group is \(4!\) So

\begin{align*} \# \{ \text{study groups} \} & = \dfrac{\# \{ \text{structured groups} \}} {\# \{ \text{equivalent groups with a given membership} \}}\\ & = \dfrac{P(20, 4)}{4!} \\ & = \dfrac{20}{4!(20 - 4)1} \text{.} \end{align*}