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Show $\large\sum\limits_{j=0}^{r}\binom{j+k-1}{k-1}=\binom{r+k}{k}$

Hint: Place $r$ balls in $m$ urns, in how many of this arrangements can you find $b$ balls in the first urn.

I'm sure that there are many other methods to solve this problem, but I have to do it with the given hint.

It is clear for $r$ balls in $m$ urns the possible number of distributions is $\large\binom{r+m-1}{m}$

Now. if I have $b$ balls in the first urn, $r-b$ balls must be distributed into the remaining urns ($m-1$). Hence $\large\binom{(r-b)+(m-1)-1}{m-1}=\large\binom{r-b+m-2}{m-1}$

and summing over b from $0$ to $r$ must give the first formula;

$\large\sum\limits_{b=0}^{r}\binom{r-b+m-2}{m-1}\overset{!}=\binom{r+m-1}{m}$

Now I'm stuck. From this formula, I can't deduce the required identity, Is there a chance to transform it ? Thanks in advance.

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1 Answers1

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For $r$ balls in $m$ urns the number of possible distributions is $\left( \array{ r + m - 1 \\ m - 1 } \right)$, not $\left( \array{ r + m - 1 \\ m } \right)$. So we see that on the left side of the equation, $\left( \array{ j + k - 1 \\ k - 1 } \right)$ is the number of ways you can distribute $j$ balls into $k$ urns, and on the right side, $\left( \array{ r + k \\ k } \right)$ is the number of ways you can distribute $r$ balls into $k + 1$ urns.

Your instinct that you should be summing $b$ over the range $0$ to $r$ was a good one, but due to the error in the formula for $r$ balls in $m$ urns, it seems you were led into trying to set $m = k$. Using the correct formula, $m = k + 1$ becomes a more obvious tactic. You have set up enough parts of the proof already that I think this will enable you to complete it.

David K
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