This video is a great introduction to permutations, combinations, and constructive counting: https://www.youtube.com/watch?v=t6a4uHEwQnM&
A given permutation of a finite set can be denoted in a variety of ways. The most straightforward representation is simply to write down what the permutation looks like. For example, the permutations of the set are and . We often drop the brackets and commas, so the permutation would just be represented by .
Another common notation is cycle notation.
The Symmetric Group
A permutation in which no object remains in the same place it started is called a derangement.
Picking Ordered Subsets of a Set
An important question is how many ways to pick an -element subset of a set with elements, where order matters. To find how many ways we can do this, note that for the first of the elements, we have different objects we can choose from. For the second element, there are objects we can choose, for the third, and so on. In general, the number of ways to permute objects from a set of is given by .