Guided Learning Extension
A circular permutation is a way to order people or objects in a circle, rather than in a row. Examples would be how to order animals on a merry-go-round or items on a lazy susan. The formula for a circular permutation is different than a permutation because there isn't a distinct beginning or end. For instance, if we wanted know how many ways we could arrange 3 people around a circular table, we might sketch a diagram with these possibilities. The arrangements are made by putting each person at the top of the table and changing the order of the other two.
A circular permutation is a way to order people or objects in a circle, rather than in a row. Examples would be how to order animals on a merry-go-round or items on a lazy susan. The formula for a circular permutation is different than a permutation because there isn't a distinct beginning or end. For instance, if we wanted know how many ways we could arrange 3 people around a circular table, we might sketch a diagram with these possibilities. The arrangements are made by putting each person at the top of the table and changing the order of the other two.
Upon further inspection, we can observe that there are only \(2\) distinct permutations because each arrangement actually has two others that are exactly the same arrangement.
The formula for circular permutations is: \(\large{\frac{n!}{n}}\ \text{or}\ (n-1)!\).
Example 1: How many ways can you arrange \(4\) people around a circular table?
\(\large{\frac{4!}{4}}\)
\(\large{\frac{4\cdot3\cdot2\cdot1}{4}}\)
\(3\cdot2\cdot1\)
\(6\)
A key ring permutation is similar to a circular permutation, except that there are half as many arrangements because you can observe a key ring forwards and backwards. Notice the example below. You can drag each colored dot on the left hand key ring, and see what the permutation would look like from the other side of the key ring. Try dragging the colored dots to make different permutations. How many distinct permutations can you make?
You can only make \(1\) distinct permutation with \(3\) items on a key ring.
Key ring permutation formula: \(\large{\frac{\left(n-1\right)!}{2}}\)
Example 2: How many ways can you arrange \(4\) of \(6\) charms on a charm bracelet?
First, figure out how many combinations of \(4\) items taken from \(6\) items there are.
\(\large{_6C_4}\)
\(15\) combinations
Then figure out how many ways there are to arrange \(4\) items on a charm bracelet.
\(\large{\frac{\left(4-1\right)!}{2}}\)
\(3\) ways
Multiple the number of combinations by the number of ways. \(15\cdot3=45\) There are \(45\) ways to arrange \(4\) of \(6\) charms on a bracelet.
Guided Learning