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Math Help - Choosing pairs of numbers with a sum-bound

  1. #1
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    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
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  2. #2
    Member courteous's Avatar
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    Let's denote what you called a "constraint" as n. Now, you can only make pairs of numbers a_1 and a_2, such that a_1+a_2<n, when a_1,a_2\in [1,n-2]. Therefore, there are n-2 pairs you can make with 1, n-3 pairs with 2, n-4 pairs with 3, ... , and 1 pair with the (n-2) number: (n-2)+(n-3)+...+2+1=\frac{(n-2)(n-1)}{2} is the numbers of pairs you seek.
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