I notice that the smallest possible difference is 1, which can happen in several ways, such as 2 - 1, 3 - 2, 4 - 3, 5 - 4, and 6 - 5. The largest possible difference is 5, which can only happen in one way, 6 - 1. I wonder if there are any gaps between 1 and 5 that cannot be represented as a difference of two distinct members of the set. I try some examples. 2 can be represented as 3 - 1 or 4 - 2. 3 can be represented as 4 - 1, 5 - 2, or 6 - 3. 4 can be represented as 5 - 1 or 6 - 2. I see that every integer between 1 and 5, inclusive, can be represented as a difference of two distinct members of the set. Therefore, the answer is 5, since there are 5 different positive integers that can be represented as a difference of two distinct members of the set. 
How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, 4, 5, 6 \}$?