Text ExplanationWe know that the average of seven different numbers is 5. The word "different" implies that no number appears twice on the list. For example, having a set in which all the entries equals 5 would be excluded by this restriction.
Certainly, we could have a symmetrical set of consecutive integers, centered on 5. This would be the set {2, 3, 4, 5, 6, 7, 8}. Both the mean and the median of this set are 5. This choice for the set makes the columns equal, suggesting the answer (C) for at least this choice.
If we can select another set that fits the criteria, mean = 5, and makes one column bigger than the other, then we would have chosen two different possibilities that give two different relationships, and BAM! right away we would know that (D) would be the answer. Can we find another set that fits the requirements?
Well, we could subtract one from each of the first six numbers, to get the consecutive integers 1-6. To keep the same average and same sum, we would have to add six to the highest number, 8 + 6 = 14. This gives us another set with an average of 5, the set {1, 2, 3, 4, 5, 6, 14}. This set has a median of 4, which is less than the 5. This choice suggests column (B) as a relationship.
Two different choices, two different relationships between the columns. BAM! This means that the answer has to be (D).
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