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Response Details:
I assume we are talking about functions f from X to X. Since X is finite, any injection is also a surjection and vice versa, so in each of the first cases we are looking at bijections on X. There are 4! = 24 such mappings, since there are 4 choices for where 1 goes, then 3 for where 2 goes, etc.
For total orderings, since we are looking at a finite set, there is a least element, a next least element, etc. Thus, there are 4 choices for the least element, 3 for the next, etc., giving 4! = 24 such orderings.
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