Number of ways to put $k$ balls in $n$ boxes

Question

Compute the number of ways to spread $k$ identical balls over $n$ different cells (where $k \geq n$) with the condition that every cell will have at least one ball.

So if $k=n$ then, we have only one option, because they're identical. otherwise, it was k! options.

If $k>n$ then, we will put one ball to every cell and will have left with $k-n$ identical balls. then it is easy. Spreading k-n balls to n different cells without any conditions is $C(k-n + n - 1, k-n - 1)$.

What do you guys think?

I believe this is how I would do it, too: reduce it to a problem of $k-n$ balls in $n$ cells, just like you did. — David K, Dec 18 '15 at 20:12

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

1

This is a standard stars and bars problem.

We have $n$ cells, they can be separated by $n-1$ bars. Our $k$ balls are the "stars". However only $n-k$ can be moved freely. So we have $n-1$ bars and $k-n$ stars. They can be arranged in $\binom{k-1}{n-1}$ ways.

Final answer is $\binom{k-1}{n-1}$. You can find a more complete explanation here

edited Apr 13 '17 at 12:21

Community

1

answered Dec 18 '15 at 19:26

Asinomás

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Number of ways to put $k$ balls in $n$ boxes

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