angel2009 wrote:

If the mean of set S does not exceed mean of any subset of set S , which of the following must be true about set S ?

I. Set S contains only one element

II. All elements in set S are equal

III. The median of set S equals the mean of set S

A. none of the three qualities is necessary

B. II only

C. III only

D. II and III only

E. I, II, and III

If the mean of set S does not exceed the mean of any subset of set S, then all the elements in S must be equal. That is because if there is an element that is not equal to the others, then the mean of S will exceed the mean of at least one subset of S. For example, if S = {1, 4, 4}, we see that the mean of S is 3, and the mean of a subset of S, say {1, 4}, is only 2.5.

Thus, we see that Roman numeral II is true. Furthermore, if all the elements in S are equal, then the median of S equals the mean of S, so Roman numeral III is true also.

Answer: D

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