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Subsection 4.1.5 Division Rule
Theorem 4.1.5.1 . Division Rule.
Suppose we can complete a task using a procedure that can be carried out in
\(n\) ways, and for each of these ways exactly
\(d\) of the ways corresponds to the same way. Then there are
\(\frac{n}{d}\) ways to do the task.
Example 4.1.5.2 . Seating arrangements.
Seat four people at a circular table.
We consider two seating arrangements to be the same if each person has the same left and right neighbor. In other words, two seating arrangements are the same if one can be rotated into the other.
How many seating arrangements are there ?
Solution .
By the Product Rule there are
\(4\cdot 3 \cdot 2 \cdot 1 = 24\) ways to arrange the four people around the table. For each way, exactly four give the same seating arrangement. By the Division Rule, there are
\(\frac{24}{4} = 6\) different seating arrangements.
