Combinatorics: arranging $n$ balls in $k$ cell with condition

Asked May 25 '19 at 10:54

Active May 25 '19 at 11:13

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Im learning combinatorics, and I came across a question I couldn't find the answer to:

I have $n$ identical balls and $k$ different cells, I want to find the number of ways to arrange the balls with the condition that each cell can have at most $x$ balls where $0\le x<n$.

Without the condition it is just arranging of $n$ balls into $k$ cells and the solution is just $\binom{n+k-1}{k}$

Can someone explain to me how we can add the condition and find the number of permutations.

Thanks

edited May 25 '19 at 11:13

asked May 25 '19 at 10:54

David

1

You could use the Inclusion-Exclusion Principle or find the coefficient of $t^n$ in the expansion of the generating function $(1 + t + t^2 + \cdots + t^x)^k$. – N. F. Taussig May 25 '19 at 14:36
@N.F.Taussig I couldn't understand how to use it, can you answer, with the guidelines please? – David May 25 '19 at 16:15

Combinatorics: arranging $n$ balls in $k$ cell with condition

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