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I want to distribute n indistinguishable objects to k containers with s0,s1,...,sk sizes. The order of the objects and the distrubution in one container doens't matter, only their amount counts. I tried several things with binomial and multinomial coefficients, be were unable to solve the problem. Any help or ideas are appreciated.

Here an example to clarify: You have 7 objects and 2 containers with sizes of 5 and 3. It is possible to put 5 objects in the first and 2 objects in the second container. It is also possible to put 4 objects in the first and 3 objects in the second container. So the possible distributions are 2.

kaetzacoatl
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1 Answers1

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Let $m_i\leq si$ , $i=0,1,2,\dots,k$ and $\sum_im_i=n $. Choose $m_0$ objects from $n$ objects and put them in the container $s_0$ which of size $s0$, and choose $m_1$ objects from $n-m_0$ objects and put them in the container $s_1$,..., and choose $m_k$ objects from $n-m_0-m_1-\ldots -m_{k-1}$ objects and put them in the container $s_k$. This mean is

$C(n,m_0). C(n-m_0,m_1)\ldots C(n-m_0-m_1-\ldots -m_{k-1},m_k)=\frac{n!}{m_0!.m_1!.\ldots m_k!}$

So the answer is $\sum_{m_i\leq si,\sum_im_i=n }\frac{n!}{m_0!.m_1!.\ldots m_k!}$

user198613
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