# How to create a formula to calculate the different combinations of money amount distribution?

Let's say that you have an amount of 100 euros and you want to distribute all of it between 3 persons without taking into consideration the decimal fractions of the euro.

If we started with some simple scenarios, then we could say:

If instead of 3 we had just one person and since we have to distribute all of the amount then we would have just a single case.

If we had 2 persons then, I guess, that we would have 100 different combinations, of distribution among the persons.

For let's say 3 persons, I have tried to solve it with Binomial coefficient, since we are talking only about integers, like this:

C(n,r) = C(100,3) = (100!)/(3!(100−3)!) = 161.700


But the result seems pretty high and I guess that this is probably because this formula takes into consideration that every n is different, or something like that.

However, in a problem where n is currency unit and there is also the rule of distributing all of the amount, which means that we want to exclude the combinations like
1€-1€-1€,
1€-1€-2€,
... then to be honest, I do not know how to approach this problem to create a formula, so any ideas would be much appreciated.

• Google "stars and bars." Jul 6 '19 at 15:47
• @saulspatz Doesn't "stars and bars" lead to a binomial coefficient? Which again is solved using nCr ?
– user634882
Jul 6 '19 at 15:50
• Yes, of course. I was responding to the last line, which I take to be the question. Jul 6 '19 at 15:52
• Are you assuming that each person receives at least some of the money? Jul 6 '19 at 15:58
• @N.F.Taussig I would like to be able to calculate both of the cases. For example to be able to determine the minimum amount of each person.
– user634882
Jul 6 '19 at 16:03

Let "o" represent $$1$$ Euro and let "|" represent divisor. It is obvious that we need $$2$$ divisors and applying repeated permutation into "ooo...o||" ($$100$$ "o" and $$2$$ divisors) gives us $$\binom{102}{2}=5151$$ different distribution. $$\\$$ For example, "ooo...o|ooo...|ooo" ($$51$$ "o" on left, $$46$$ "o" on middle, $$3$$ "o" on right) is $$51$$ Euro for $$1^{st}$$ person, $$46$$ Euro for $$2^{nd}$$ person and $$3$$ Euro for $$3^{rd}$$ person.
• If each person gets X minimum value, then the result will be $\binom{102-3x}{2}$ (first give each one in X amount, then use stars and bars) Jul 6 '19 at 21:07