Circular permutations without numbered positions For n distinct objects in a circle, rotations of one arrangement are equivalent. Each circular arrangement corresponds to n linear arrangements, so divide n! by n. (n!)/(n)=(n-1)! Circular permutations with numbered positions When the positions are numbered, every position is distinct. A circular arrangement is then equivalent to a row arrangement. (n-1)!,n=n! Miscellaneous counting problems Apply the addition and multiplication principles flexibly. Common ideas include: - deciding whether the order of objects must be included; - splitting the count into different cases and adding; - splitting the task into different steps and multiplying; and - excluding repeated counting where necessary.

Circular permutations without numbered positions

For \(n\) distinct objects in a circle, rotations of one arrangement are equivalent. Each circular arrangement corresponds to \(n\) linear arrangements, so divide \(n!\) by \(n\).

Unnumbered circle

Circular permutations with numbered positions

When the positions are numbered, every position is distinct. A circular arrangement is then equivalent to a row arrangement.

Numbered positions

Apply the addition and multiplication principles flexibly. Common ideas include:

deciding whether the order of objects must be included;

splitting the count into different cases and adding;

splitting the task into different steps and multiplying; and

excluding repeated counting where necessary.

Circular Permutations and Counting Strategies

Circular Permutations and Counting Strategies