Let $A$ be a set of $n$ elements. Let $S\_1,\dots,S\_n$ be independent $k$-element random subsets of $A$. What is the probability that $S\_1,\dots, S\_n$ evenly cover $A$, i.e. each element of $A$ belongs to exactly $k$ random subsets?



Random covering of a set
It follows immediately from the special case (with $m=n$ and $s=t=k$) of Theorem 1 of [Canfield and Mckay](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29) that the probability -- say $p\_{n,k}$ -- in question is
$$\sim\frac1{\sqrt e}\dfrac{\displaystyle{\binom{n}{k}^n}}{\displaystyle{\binom{n^2}{k n}}} \tag{1}\label{1}$$
if $n\to\infty$ and either $k=o(n^{1/2})$ or $k\asymp n-k\asymp n$. (Indeed, writing $M\_{ij}:=1(j\in S\_i)$ for all $i$ and $j$ in $[n]:=\{1,\dots,n\}$, we have a bijection between the set of all "even" $k$-coverings $(S\_1,\dots,S\_n)$ of the set $[n]$ and the set of all $0$-$1$ "incidence" matrices $M=(M\_{ij}\colon(i,j)\in[n]^2)$ with all row sums and all column sums equal $k$.)


According to Conjecture 1 of [Canfield and Mckay](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29), \eqref{1} holds uniformly over all $k\in\{1,\dots,n-1\}$ as $n\to\infty$.


A prehistory of this problem is discussed in [Canfield and Mckay](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29) as well.




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For $k=2,3,4$,
$$p\_{n,k}=C\_{n,k}\Big/\binom nk^n,$$
where $(C\_{n,2})$, $(C\_{n,3})$, and $(C\_{n,3})$ are [A001499](https://oeis.org/A001499), [A001501](https://oeis.org/A001501), and [A058528](https://oeis.org/A058528), respectively.


