It would be helpful if we could see the exact wording of the original question, since this one, as you have punctuated it, does not have a very clear meaning.
Although I have some thoughts which may be helpful, I can't see quite where the comes from. So I am open to corrections or further suggestions/information. My thoughts are these:
The number of ordered pairs, , that can be formed from the elements of the set S is .
Each element is an ordered sequence made up of elements of S. The number of ordered pairs , that can be made from elements of S is . So (if I am understanding the question correctly) this is the number of ordered pairs of that each element will 'consume', if each ordered pair can be used at most once in this way. So it would appear that a maximum value on the number of possible elements is:
This appears to show that , not the of which the question speaks.
Am I misinterpreting something? Is there something wrong with my reasoning?