permutation

A permutation of a set of objects is an ordering of those objects. For instance, the possible permutations of the set \(\{a,b,c\}\) are \(abc\), \(acb\), \(bac\), \(bca\), \(cab\), and \(cba\).

The number of distinct permutations of all the elements of a set of \(n\) elements is given by \(n!\) (\(n\) factorial), i.e. the product

\[n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1.\]

More generally, the number of permutations of a subset of \(r\) elements chosen from a set of \(n\) elements is given by the permutations formula

\[\displaystyle\frac{n!}{(n-r)!},\]

generally denoted by \(_nP_r\) and described as “\(n\) permute \(r\).”