“Stuffing some stuff into buckets”

How many ways are there to sort \(n\) distinct objects to \(r\) buckets?

\begin{equation} r^{n} \end{equation}

## grouping with entirely indistinct objects

You can simply reframe the grouping problem as permutation of the objects with \(r-1\) dividers along with your old \(n\) objects.

i.e.: sort this thing —

So:

\begin{equation} \frac{(n+r-1)!}{n! (r-1)!} \end{equation}