Final answer:
Given the mode of 4 and median of 3, one possible set of four positive integers that satisfies these conditions and a valid answer choice is {2, 3, 4, 4, 5} with a sum of 18.
Step-by-step explanation:
The question involves finding the sum of four positive integers given that they have a mode of 4 and a median of 3. The mode being 4 implies that 4 appears at least twice. The median of 3 means that the second and third numbers in the ordered list are either both 3 or the numbers immediately above and below 3, for example, 2 and 4.
To find the possible integers, let's consider the mode and median. Since the mode is 4, we have at least two 4's. For the median to be 3, we could have the following numbers: 2, 3, 4, 4. The sum of these numbers is 13. However, we are looking for four positive integers that satisfy the given conditions and result in one of the provided sum options (18, 19, 20, or 21).
We need to add another number to the set to make the sum equal one of the provided options. If we add the least possible positive integer, which is 1, we get the set {1, 2, 3, 4, 4}. The sum of this set is 14, which is not one of the options. If we add 5 instead, the set becomes {2, 3, 4, 4, 5} with a sum of 18, which is one of the given options a) 18. This set fits the conditions for mode and median and also corresponds to one of the answer choices.