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I have the following combinatoric problem :

I want to distribute n (equivalent) atoms among M distinct objects. Each object can contain from 0 to nlim atoms. How many combination do I have for this system?

If nlim>n, this problem actually corresponds to the classical "Number of combinations with repetition". But in my case nlim<n. In fact, I am interested in the limit of (lnΩ)/n (Ω beeing the number of combination) when M and n tend toward infinity (with n=a M where a is a constant) while nlim is finite (and actually rather small)..

I found a solution for that problem using some series of summations but it will be impossible to caculate as soon as M and n become large (even for M=100, n=300 and nmax=10, it took my laptop more than one hour to solve it).

Is there a simple analytical solution to this problem?

Thank you for your help.

Emile Maras

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# Number of combinations with limited repetition

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