For a lot of average problems, it is helpful to think of the "balanced scale".
We know the average of all the numbers is 4. In the set, we see the number "2". This number takes us 2 to the left of the average of 4. Each "3" takes us 1 to the left. The "8" takes us 4 to the right. To sum up:
LEFT-----------RIGHT
-2----------------+4
-1
-1
We have -4 on the left, and +4 on the right. Thus, the numbers on the left balance out the ones on the right. Thus, K and M cannot collectively deviate from the set's average of 4 (not without changing the average). Thus, the average of K and M must be 4. Thus, their sum must be 8. Because we are told that K does not equal M, one possible solution set is 5 and 3 (two unequal numbers that add to 8).
Let's observe the median if K and M are 5 and 3:
{2, 3, 3, 3, 5, 8}
We have an even number of numbers in the set (ie, 6). Thus, the median is the average of the middle 2. Clearly, here the median is 3.
Because there can be only one correct answer, and because we don't have "it cannot be determined" as one of our answer choices, the answer must be "3", and there is no need to consider other solution sets (6 and 2, 7 and 1, 8 and 0, etc.).
Choose C.
Kaplan Teacher in Toronto