A permutation is an arrangement of objects in which the order is important. For example, consider constructing a number using only the digits $4,$ $5,$ and $6$ without repetitions. Any of the three digits can be picked for the first position, leaving two choices for the second position, then only one choice for the third position. In this case, there are six possible permutations. Although all these numbers are formed with the same three digits, the order in which the digits appear affects the number produced. Each different order of the digits creates a different number. The number of permutations can be calculated by using the Fundamental Counting Principle.

Listing all the permutations may be a difficult task when many objects are being arranged. In these cases, the Permutation Formula can be used instead.

Circular Permutation

In a circular permutation, a set of objects is arranged in a circle or loop with no fixed reference point. In this case, arrangements in which the objects have the same order relative to each other are indistinguishable. Consider a group of people sitting in a circular table. Rotating the table does not change the order of the people sitting relative to each other. Ali is always sitting next to Magdalena, Dominika is always sitting next to Ali, and so on.

Permutation With Repetitions

In permutations with repetitions, there are indistinguishable objects in the set to arrange. Therefore, some arrangements cannot be distinguished from each other. Consider all the possible arrangements that can be made with one digit $0$ and two digits $1.$ To distinguish between different arrangements, the digits $1$ will be written as $1$ and $1.$ If the colors are removed, there are only three distinguishable arrangements.