To find the possible differences, I need to subtract every pair of numbers in the set, not counting order. For example, 1 - 3 and 3 - 1 are the same difference, so I only count it once. I also don't need to subtract a number from itself, since that would give zero, which is not positive. So I have to consider 21 pairs of numbers, which are the combinations of 7 numbers taken 2 at a time. I can list them in a systematic way, starting from the smallest number and going up. 1 - 3, 1 - 5, 1 - 7, 1 - 9, 1 - 11, 1 - 13, 3 - 5, 3 - 7, 3 - 9, 3 - 11, 3 - 13, 5 - 7, 5 - 9, 5 - 11, 5 - 13, 7 - 9, 7 - 11, 7 - 13, 9 - 11, 9 - 13, 11 - 13. I can also calculate the differences for each pair. -2, -4, -6, -8, -10, -12, -2, -4, -6, -8, -10, -2, -4, -6, -8, -2, -4, -6, -2, -4, -2. Since I only care about the absolute values of the differences, I can ignore the negative signs. So the possible differences are 2, 4, 6, 8, 10, and 12. These are all distinct, so there are 6 distinct positive integers that can be represented as the difference of two numbers in the set. 
How many distinct positive integers can be represented as the difference of two numbers in the set $\{1, 3, 5, 7, 9, 11, 13\}$?