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I am stucked at this problem:

It is known that the number of ways to arrange $k$ non-distinguished balls into $n$ cells is $\binom{n+k-1}{ k}$, Now by partitioning the cells into disjoint subsets, We can express $\binom{n+k-1}{ k}$ as a sum of the form $\Sigma_{i=0}^k a_i$.

Find the required sum (I.e. Find $a_i$).

Thanks for any hint/help.

Martin Sleziak
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MathNerd
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    I think that any lazy mathematician (as I am) would write this sum as $0+ 0+ 0+ \dots + 0 + \binom{n+k-1}{k}$. – Crostul Jun 01 '15 at 09:11
  • @Crostul Yes, but it says that by partitioning the cells into disjoint sets we can get a sum that has a combinatorial meaning. – MathNerd Jun 01 '15 at 09:21
  • Almost a duplicate of http://math.stackexchange.com/questions/833451/prove-sum-i-0n-binomik-1k-1-binomnkk – Jack D'Aurizio Jun 01 '15 at 10:42

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HINT: Partition the cells into two blocks, one containing the first $n-1$ cells and one containing only the $n$-th cell. For $i=0,\ldots,k$ let $a_i$ be the number of ways to distribute the $k$ balls so that the $n$-th cell contains $i$ balls; $a_i$ is then the number of ways to distribute $k-i$ balls amongst the $n-1$ cells of the first block.

(Unlike Lucian, I’m assuming that you stated the problem correctly, i.e., that the upper limit of summation really is $k$, not $n$.)

Brian M. Scott
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