Distributing infinite supply of $n$ distinct objects into $k$ identical urns I have $n$ **distinct** objects, namely {$n_{1\le i \le n}$} with an **infinite** supply of each of them, and I have $k$ **identical** , indistinguishable urns to place the objects in. Each urn will contain **exactly** one item. Let $P(n,k)$ be the number of ways to do it. $P(2,2)=3$. There are 3 different ways to do it: {$n_1,n_1$}, {$n_2,n_2$}, {$n_1,n_2$} $P(3,3)=10$ because: {$n_1,n_1,n_1$}, {$n_2,n_2,n_2$}, {$n_3,n_3,n_3$},{$n_1,n_1,n_2$}, {$n_1,n_1,n_3$},{$n_2,n_2,n_1$}, {$n_2,n_2,n_3$}, {$n_3,n_3,n_1$}, {$n_3,n_3,n_2$}, {$n_1,n_2,n_3$} Note that I do not constrict myself to $n=k$ although these were my examples. What is the general formula/solution to $P(n,k)$ if such one exists? Thanks in advance.

This is a non-standard rewording of the usual Stars and Bars problem (please see Wikipedia). Let $x_i$ be the number of urns that will contain an object of Type $i$. We want to find the number of solutions in non-negative integers of the equation $x_1+\cdots+x_n=k$.

The number of solutions is $\binom{k+n-1}{n-1}$, or equivalently $\binom{k+n-1}{k}$.