When the elements of a set are rearranged in a way that the order of the elements is important, each possible arrangement is called a permutation. For example, consider constructing a number using only the digits $1,$ $2,$ and $3,$ with no repetitions.

To create a permutation, one digit needs to be chosen for the first position. This leaves $3-1=2$ digits to choose for the second position. This process repeats until the end of the permutation. In this case, there are $6$ possible permutations. $$\begin{array}{c}123132213231312321\end{array}$$ Even though all of these numbers are made up of the same three digits, the order in which they appear changes the value of each number. The number of permutations can be calculated by using the Fundamental Counting Principle.

This process can be generalized resulting in the Permutation Formula, which calculates the number of permutations that can be formed by arranging $n$ elements of a set in ordered groups of $r$ elements. Note that the exclamation points in the formula indicate that the factorial of the value should be calculated.

In the previous example, there were $n=3$ options for the digits to use to make groups of $r=3$ — in this case each group was a $3\text{-}$digit number.

An alternate notation for ${}_n{P}_r$ is $P(n,r).$