I need to find the probability of calculating placing of r balls into n urns, such that there can be no empty urns.

I find the number of ways to do it is $\displaystyle {n-1}\choose{r-1}$.

But this is pretty useless. It does not give me the actual probability.

For example, for n=2, r=3.

We can list down all the cases:

ABC|x

x|ABC

AB|C

AC|B

CB|A

A|BC

B|AC

C|AB

The probability is 6/8.

The number of ways is 6.

Using the $\displaystyle {n-1}\choose{r-1}$ formula, I get:

$\displaystyle 1\choose2$, which is pretty much meaningless.

Can I use any mathematical models to get the number 6, instead of having to list all down?

Thanks!