Wikipedia: Permutations

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A permutation, in combinatorics, is a sequence of elements in which no element appears twice. In a sequence, unlike in a set, the order in which the elements are written down matters. Suppose you have a total of n distinct objects at your disposal and you want to create permutations of k elements selected from those n, where k≤n. In how many ways can that be done?

We can select the first member of the list in n ways because there are n distinct elements.
The second member of the list can be filled in (n-1) ways since we have used up one of the n elements already.
The third member can be filled in (n-2) ways since 2 have been used already.
This pattern continues until there are k names on the list. This means that the last member can be filled in (n-k+1)'' ways.

Summarizing, we find that a total of

: n * (n-1) * (n-2) * ... * (n-k+1)

different permutations of k objects, taken from a pool of n objects, exist. If we denote this number by _nP_k and use the factorial notation, we can write

: _nP_k = n! / (n-k)!

In abstract algebra and other fields, the term permutation is usually reserved for a bijective map from a finite set to itself. There are two main notations for such permutations:

one can just arrange the "natural" ordering of the elements being permuted on a row, and the new ordering on another row:

1 2 3 4 5 2 5 4 3 1

or one can write the cycles induced by the permutation; that is, one takes any element (say 1); then the element the first one is being sent to (here 2); then the element this one is sent to (here 5), and so on, until one comes back to the first. This is a cycle. The next cycle begins with any other element not considered till now, until every element appears in a cycle. So the previous permutation has cycle form (1 2 5)(3 4). Of course, the same permutation could be written as (4 3)(2 5 1), or any other variant.

Two permutations of a set of n elements (often, {1, 2, 3, ..., n}) can be composed, i.e. applied successively to the set. For instance, if a = (125)(34), and b = (13)(2)(45), applying b after a maps 1 to 2, and then to itself; 2 to 5 to 4; 3 to 4 to 5, and so on. So composing b and a gives ba = (124)(35).

Under this operation of composition, all the permutations of a set, or a suitable set of them, form a mathematical group, called a permutation group.