Set B has three positive integers with a median of 9. If the largest

I selected answer as B because it is the ONLY option among the given numbers as answer choices that do satisfy the condition.

Makes sense,clearly I misunderstood the question.

What is the source? The question makes no sense as written. If a GMAT PS question tells you "x is an integer", then x is some fixed integer that we don't know. The question can't then talk about "the largest possible value of x", because x is a fixed number. It's not something that can vary, so it can't have a maximum or minimum value. The same is true here: if B is a set of three integers, B is some fixed set that has some fixed range. It makes no sense to talk about its "largest possible range", because set B is not a variable. Its range has one value and one value only.

The question is trying to ask something a bit convoluted, something the wording doesn't correctly convey. It is trying to ask: suppose you were told to make a set of three positive integers, and you were told the set needs to have a mean equal to some number M and a median of 9. If you then discovered that 19 was the largest possible range this set could have, what is M?

To answer that question: if we know the mean of the set, then we know the sum of the three values in the set (the sum is the mean multiplied by 3). Since we know the middle value is 9, we could then find the sum L + S of the largest (L) and smallest (S) values. So we could find the value of L+S, and if we want to maximize the range L-S, we want to make S and L as far apart as possible, so we want to make S as small as we can and L as large as we can. But S is a positive integer, so S = 1 is its minimum value. So if we figured out that the maximum possible range of our set is 19, then since we'd always make S = 1 to maximize that range, we'd then know L = S + 19 = 20, so the set needs to be 1, 9, 20, and the mean is 10.