We'll think about this question logically.
If our set is, for example, 2 4 6 8, the mean of the set is 5. If we take subset 2 8 or 4 6, the mean will still be 5. However, if we take subset 2 4 6, the mean will be 4. If we take subset 4 6 8, the mean will be 6. We should note that if our subset only eliminates numbers above the mean, we will get a subset mean that is less than the original mean; if our subset only eliminates numbers below the mean, we will get a subset mean that is greater than the original mean.
We want to find a set where the mean of every possible subset is the same as the mean of the set. If the numbers in our set are different, we will always be able to pick a subset that eliminates more of the higher numbers or more of the lower numbers, yielding a smaller or larger mean. However, if all of the numbers in our set are the same, we can eliminate as many of them as we'd like, and our mean will be the same. For example, in set 4 4 4 4 4, our mean is 4. If we take subset 4 4 4, our mean is still 4.
Knowing that, we'll look at our options.
I. If the original set contains only one element, all subsets will be the same as the original set, which will result in the same mean. However, we established that this doesn't *need* to be true - we can have multiple elements, as long as they have the same value. We can eliminate answer choice E.
II. This is exactly what we stated must be true. We can eliminate A and C.
III. If all numbers in a set or subset are the same, the median will have the same value as the numbers. For instance, in our hypothetical 4 4 4 4 4 set AND our 4 4 4 subset, the median is 4. So since all of the numbers in the set and all of its possible subsets must be the same, the median must also be the same. This must be true, so the correct answer is II and III, or D.
