If x is the mode of a set with 5 elements and x is less than exactly

Since x is the mode of a set with 5 elements, there must be at least 2 elements in the set that are x. Since x is less than exactly two members of the set, there must be either 2 elements or 3 elements in the set that are x. Therefore, the possible scenarios for the 5 elements (from smallest to largest) are (note: we are assuming each variable below has a unique value):

1) w, x, x, y, z
2) x, x, x, y, z
3) x, x, x, y, y

Now, let’s analyze each Roman numeral.

I. x is the median of the set.

We can see that in all 3 possible scenarios, x is the median of the set. Roman numeral I is true.

II. The median of the set is greater than the mean of the set.

Since we don’t know actual values of the 5 elements in the set, Roman numeral II might be or might not true. For instance, if we use scenario 3 above and x = 5 and y = 10, the median = 5, while the mean is (5+5+5+10+10)/5 = 35/5 = 7.

III. x is less than the greatest member of the set.

Looking at our 3 scenarios, we see that x is always less than the greatest member of the set. Roman numeral III is true.

Answer: D