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The number of ways to put $n$ unlabeled balls in $k$ distinct bins is $\binom{n+k-1}{k-1} .$ Which makes sense to me, but what I can't figure out is how to modify this formula if each bucket has a max of $m$ balls.

EDIT: What I've tried:

I got to the generating function $(1-x^{m+1})^k(1-x)^{-k}$ which ends up giving me $\sum_{r(m+1)+r_2=n} \binom{k}{r}(-1)^{r_2}\binom{k+r_2-1}{r_2}$

But when programing this:

def distribute_max(total,buckets,mmax):   ret = 0   for r in xrange(total//(mmax+1)+1):     r_2 = total - r*(mmax+1)     ret += choose(buckets,r) * (-1)**r_2 * choose(buckets + r_2 - 1,r_2)   return ret 

I'm getting terribly wrong answers. Not sure which step I screwed up.

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    Thanks @naktinis. Fixed link: complete derivation [here](http://cogcomp.org/~mayhew2/stackexchange.pdf).2018-10-02

3 Answers 3

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As a check I did it with an inclusion-exclusion argument, getting $\sum_i(-1)^i\binom{k}i\binom{n+k-1-i(m+1)}{k-1}\,.$

Setting $r=i$ and $r_2=n-i(m+1)$ to match your notation, I make this $\sum_r (-1)^r\binom{k}r\binom{k+r_2-1}{r_2}\,.$ It appears that you’ve the wrong exponent on $-1$.

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Let's denote the problem as $D(n,k,m)$, then when $mk-n \leq m$, the answer can be given as: $D(n,k,m)=\binom{km-n+k-1}{k-1}$, where $m \leq n$.

It can be easily verified using $D(5,2,3)=2$, or $D(6,3,3)=10$, etc..

Let me try to explain it in more details with my poor English (sorry for that).

Suppose we start from the state that all bins are filled with $m$ balls in each. Then the task for us is to eliminate $mk-n$ balls from these $k$ bins. To ditribute the $mk-n$ "elimination" into $k$ bins, we have $\binom{km-n+k-1}{k-1}$ different ways if $mk-n \leq m$, i.e. the number of elimination in each bin will not exceed $m$.

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    Why don't you write down the solution more explicitly?2015-03-16
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The number of ways to put $n$ balls in $k$ boxes with in each box a maximum of $m$ balls is the coefficient of $[q^n]$ in the $\text{QBinomial}(k+m,k,q)$.

For example, the number of different ways to put $n = 5$ balls in $k = 4$ boxes with in each box no more than $m = 3$ balls is given by
$ \text{QBinomial}(7,4,q) = (q^6+q^5+q^4+q^3+q^2+q+1)(q^2-q+1)(q^4+q^3+q^2+q+1) = q^{12}+q^{11}+2q^{10}+3q^9+4q^8+4q^7+5q^6+4q^5+4q^4+3q^3+2q^2+q+1. $ The coefficient of $q^5$ is $4$. Hence there are four ways. These four ways are $(2,3)$, $(1,1,3)$, $(1,2,2)$, $(1,1,1,2)$.

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    The boxes are distinct, so there are 40 ways for the n=5,k=4,m=3 example. In particular, 12 each for (2,3), (1,1,3) and (1,2,2), and 4 for (1,1,1,2). The code in the question, with the correction from the accepted answer, gives 40.2017-05-16