Q38

Let $U=\{1,2, \ldots, n\},$ where $n$ is a large positive integer greater than $1000.$ Let $k$ be a positive integer less than $n$. Let $A, B$ be subsets of $U$ with $|A|=|B|=k$ and $A \cap B=\emptyset$. We say that a permutation of $U$ separates $A$ from $B$ if one of the following is true. All members of $A$ appear in the permutation before any of the members of $B$. All members of $B$ appear in the permutation before any of the members of $A$. How many permutations of $U$ separate $A$ from $B?$