Formula to get total combination possibilities

Question

• 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

Answer 1

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.