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In how many way n identical things can be distributed among r different persons where each person may get any number of things.

My book gives following ans:

(n+r-1)C(r-1)

but I am not able to reason it out.Please help.

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1 Answer

Consider the word $O\ldots O$ with $n$ letters, one for each object. You want to place $r-1$ markers $P$ among those $O$, with the interpretation that person #1 gets all objects before the first $P$, person #2 gets all objects after the first $P$ and before the second $P$, etc. Note that what the last person gets is fixed by what the others get, so you only need $r-1$ markers. In summary, you want to know the number of permutations of $O\ldots OP\ldots P$, a word with $n+r-1$ letters, $n$ being $O$ and $r-1$ being $P$. This is $\frac{(n+r-1)!}{n!(r-1)!} = {{n+r-1} \choose {r-1}}$.

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