HMMT 2003 T5

Let $B_{k}(n)$ be the largest possible number of elements in a 2 -separable $k$-configuration of a set with $2 n$ elements $(2 \leq k \leq n)$. Find a closed-form expression (i.e. an expression not involving any sums or products with a variable number of terms) for $B_{k}(n)$.