Technically, what you are asking for are the permutations of . Consider - you have five blank slots in which to fill five distinct objects:
1 2 3 4 5
_ _ _ _ _
For the first blank, you have five options to choose from, for example,
1 2 X 4 5
3 _ _ _ _
For the next blank, you have four options, etc, all the way down to the last blank, in which you only have one option, whatever number is left over. By the Fundamental Counting Principal, you get your total number of options by multiplying all these individual options together, thus getting , as Plato has shown.
There is a simple and organized way of writing all of these down, and I'll use the 4 case to show you, since indeed 5 is too long. 4!=24, so there are 24 permutations in the set. A fourth of them start with one, a fourth start with 2, etc. In the fourth starting with one, a third have a next number of 2, a third have a next number of 3, etc. Repeat this process and you will see an easy pattern emerge:
1234 2134 3124 4123
1243 2143 3142 4132
1324 2314 3214 4213
1342 2341 3241 4231
1423 2413 3412 4312
1432 2431 3421 4321
I'll leave it to you to take care of the 5 case.