Bijection of a set in itself.
A permutation of the n objects of a set E is all ordered combinations of these n elements. Therefore, a permutation of n objects in a set E of n elements is an n-tuple formed by these elements.
If E \(= \left\{0, 1, 2, 3\right\}\), then a permutation of E could be represented by the quadruplet \(\left(1, 0, 3, 2\right)\), or the matrix: \(\begin{pmatrix} 0 & 1 & 2 & 3\\1 & 0 & 3 & 2\end{pmatrix}\).
If E \(= \left\{0, 1, 2, 3\right\}\), then a permutation of E could be represented by the quadruplet \(\left(1, 0, 3, 2\right)\), or the matrix: \(\begin{pmatrix} 0 & 1 & 2 & 3\\1 & 0 & 3 & 2\end{pmatrix}\).
We can also describe a permutation as the set of ordered pairs that form the relation. In the previous case, we would have: {(0, 1), (1, 0), (2, 3), (3, 2)}.
A permutation of a set of n elements is an arrangement of these n objects taken n at a time.
Form
The number of permutations in a set E that includes n elements is equal to n!.
Example
Consider the set E = {0, 1, 2).
Therefore, we have: n! = 3! = 6.
The set of permutations of E is {(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0)}.