# Formula to get total combination possibilities

• 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, 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 for some ideas. –  hardmath Apr 23 '12 at 18:22
@BrettFrankel sorry, I wasn't familiarized with any tags, but after some time reading I think I found some that fit ^^ –  ajax333221 Apr 23 '12 at 18:58
@ajax333221: Those tags were inapproriate -- I replaced them with combinatorics. –  joriki Apr 23 '12 at 19:25

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.