How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, 4, 5, 6 \}$?
It is clear that the maximal positive difference is $6 - 1 = 5$. Moreover, we can use 6 to ensure that we can get all positive integer differences up to 5: $6 - 5 = 1$, $6 - 4 = 2$, $6 - 3 = 3$, $6 - 2 = 4$, $6 - 1 = 5$. So, there are $\boxed{5}$ possible differences.
5
Level 2
Counting & Probability
Problem: How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, 4, 5, 6 \}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many different positive integers can be represented as a difference of two distinct members of the set {1, 2, 3, 4, 5, 6}, we first identify the possible pairs of distinct integers from the set. The set contains 6 elements, and the number of ways to choose 2 distinct elements from this set is given by the combination formula C(n, k), where n is the total number of elements and k is the number of elements to choose. Here, C(6, 2) equals 15.

Next, we calculate the differences for each pair. The difference between two distinct integers a and b, where a > b, is given by a - b. The smallest possible difference occurs when we take the smallest two distinct integers, which is 2 - 1 = 1. The largest possible difference occurs when we take the largest and smallest integers, which is 6 - 1 = 5.

Now, we systematically list the differences for all pairs:

- For 2: 2 - 1 = 1
- For 3: 3 - 1 = 2, 3 - 2 = 1
- For 4: 4 - 1 = 3, 4 - 2 = 2, 4 - 3 = 1
- For 5: 5 - 1 = 4, 5 - 2 = 3, 5 - 3 = 2, 5 - 4 = 1
- For 6: 6 - 1 = 5, 6 - 2 = 4, 6 - 3 = 3, 6 - 4 = 2, 6 - 5 = 1

Next, we compile the unique positive differences obtained from these calculations: 

1, 2, 3, 4, 5.

We observe that all integers from 1 to 5 can be represented as differences of two distinct members of the set. Therefore, the total number of different positive integers that can be represented is 5.<|END_OF_LATENT|> The final answer is $5$.