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$\begingroup$
  • There are Num guys
  • The sum of their ages is Tot
  • The Age of every one of them is Min <= Age <= Max

I want to create a formula for a program to calculate the total number of possibilities

Any ideas/links that point me to the right direction are appreciated

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    What you are asking seems to be about [partitions of a number](http://en.wikipedia.org/wiki/Partition_%28number_theory%29), and more specifically partitions of `Tot - Num*Min` with at most `Num` summands and no summand greater than `Max - Min`. Formulas for this kind of thing can be pretty hard to program, but setting up a loop to run through all of the possibilities and count them is pretty straightforward. See Henry Bottomley's online [partition calculators](http://www.btinternet.com/~se16/js/partitionstest.htm) for some ideas.2012-04-23
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    @BrettFrankel sorry, I wasn't familiarized with any tags, but after some time reading I think I found some that fit ^^2012-04-23
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    @ajax333221: Those tags were inapproriate -- I replaced them with [tag:combinatorics].2012-04-23

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You can map this to the problem of balls in bins with limited capacity by subtracting $\text{Num}\cdot\text{Min}$ from $\text{Tot}$ and then distributing the remainder into $\text{Num}$ bins with equal capacity $\text{Max}-\text{Min}$. The above page gives a formula for the case of equal capacities at the very end, which can be derived using the inclusion–exclusion principle by considering the number of bins filled to capacity.