Thread: Choosing pairs of numbers with a sum-bound

Dear Forum,

I have the following question:

How can one write up a closed formula for computing the number of possible pairs of numbers in a set, whose sum is lees than or equal to a constraint?

For example: if we have the numbers from 1 to 100 what is the number of cases when the sum of the two numbers in a pair is less than 10. Here, the 1, 100 and 10 values are not important at all, I just wanted make it easier to demonstrate the problem. Of course, I would need a so called closed formula for that, I am not sure whether it exists. (Iterative solution is unfortunately not an option).

Any help would be appreciated!


Your sincerely,

Shinkansen

Let's denote what you called a "constraint" as . Now, you can only make pairs of numbers and , such that , when . Therefore, there are pairs you can make with , pairs with , pairs with , ... , and pair with the number: is the numbers of pairs you seek.