Determining all the elements of S4? What is an easy way to determine the elements of $S_{4}$? 

While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this.
I tried to search for an answer via Google but there are no satisfying responses.

Any help is appreciated.
List all possible 4-letter words in the letters $1,2,3,4$ without repetition.

A systematic way is to list them in lexicographical order:

1234
1243
1324
1342
...
4321


A word $a_1 a_2 a_3 a_4$ corresponds to the permutation $i \mapsto a_i$ in $S_4$.
We know that every element of $S_4$ is an automorphism over ${1,2,3,4}$.
We want to count the number of elements in $S_4$ then lets start with $1$, it must be mapped to an element in ${1,2,3,4}$ - so it has 4 options, now for 2 - he has only 3 options for mapping (since $1$ already occupied an element in ${1,2,3,4}$), in the same way $3$ has $2$ options, and the mapping of $4$ is determined uniquely after determining all the other mappings.

Therefore, we finally get $4\cdot 3\cdot 2\cdot 1 = 4! = 24$ such elements.