Official Explanation:
For this problem, we have few constraints, so we are going to have to start by picking numbers. Remember that picking numbers is an excellent way to demonstrate that a single statement or the pair of statements is not sufficient. If two possible choices consistent with a statement produce different answers to the prompt question, then BAM! we know right away that the statement is not sufficient. Proving that an individual statement or two statements are sufficient takes some logical reasoning: we can't establish sufficiency with picking numbers, only the absence of sufficiency.
Remember: all we know are the four elements of the set are integers: they could be positive, zero, or negative.
Statement #1: The average of K is 3.
We will approach this with picking numbers.
Choice #1 = {3, 3, 3, 3}
This has an average of 3 and the median is 3
Choice #2 = {0, 0, 0, 12}
This also has an average of 3 but the median is 0
Right there, BAM! Two different choices consistent with statement #1 lead to two different answers to the prompt question. Statement #1, alone and by itself, has to be not sufficient.
Statement #2: The mode of K is 3.
We will approach this with picking numbers.
Choice #1 = {3, 3, 3, 3}
This has a mode of 3 and the median is 3
Choice #2 = {3, 3, 5, 23}
This has a mode of 3 but a median of 4.
Again, BAM! Two different choices consistent with statement #2 lead to two different answers to the prompt question. Statement #2 alone and by itself, has to be not sufficient.
Combined statements:
Choice #1 = {3, 3, 3, 3}
Choice #2 = {2, 3, 3, 4}
Choice #3 = {– 5, 3, 3, 11}
Those three example of sets that have both a mean of 3 and a mode of 3. All of these have medians of 3. Picking numbers here doesn't prove anything, but it does suggest a pattern of reasoning.
Certainly in the set {3, 3, 3, 3}, both statements are true and the median is 3.
We absolutely can't have three 3's and one different number, because there's no way that set could possibly have an average of 3.
We could have two 3's, and two other numbers, but think about it.
We can't have {3, 3, bigger, biggest}, because that would have an average higher than 3.
Similarly, we can have {smallest, smaller, 3, 3} because that would have an average less than 3.
If we have two 3's and two other numbers, it absolute must be the case that one is larger than three and one is smaller than three, by equal amounts, so that they average out to 3. This would have to be:
{smaller, 3, 3, bigger}
This kind of set always has a median of 3.
Thus, for all possible cases consistent with the combined statements, the only possible answer to the prompt question is 3. Combined, the statements produce a definitive answer, so they are sufficient together.
Answer = (C)
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