We are given that the average of five distinct positive integers is 10. This implies that the sum of these integers is: $\text{Sum} = 5 \times 10 = 50$ We need to find the difference between the largest and least possible values of the greatest of the five integers.

Step 1: Minimize the largest integer

To minimize the largest integer, we need to make the other four integers as large as possible while still keeping the sum 50 and ensuring all integers are distinct.

Start by choosing the smallest four distinct integers: $1, 2, 3, 4$ The sum of these four integers is: $1 + 2 + 3 + 4 = 10$ Now, subtract this from the total sum of 50: $50 - 10 = 40$ Thus, the smallest possible values for the first four integers are $1, 2, 3, 4$, and the largest integer in this case is $40$.

Step 2: Maximize the largest integer

To maximize the largest integer, we need to minimize the other four integers. Start by choosing the smallest four distinct integers as large as possible while still keeping the total sum 50.

The largest possible values for the other four integers are the next smallest ones before the greatest integer: $9, 8, 7, 6$. The sum of these integers is: $9 + 8 + 7 + 6 = 30$ Now, subtract this from the total sum of 50: $50 - 30 = 20$ Thus, the greatest integer in this case is $20$.

Step 3: Calculate the difference

The smallest possible value for the greatest integer is $20$, and the largest possible value is $40$. The difference between these values is: $40 - 20 = 20$

Thus, the difference between the largest and least possible values of the greatest of the five integers is $\boxed{20}$.

Would you like more help on any of the steps? Here are 5 related questions:

1. How do you find the average of a set of numbers?
2. What are some strategies for finding the maximum and minimum values in similar problems?
3. How would the solution change if the integers were not distinct?
4. Can you solve for the median in similar problems?
5. What happens to the greatest integer if the sum is increased?

Tip: When dealing with sums, minimizing the number of large integers can help reduce the maximum value of a set!