P16095 [ICPC 2019 NAIPC] Piece of Cake

Alice received a cake for her birthday! Her cake can be described by a convex polygon with $n$ vertices. No three vertices are collinear. Alice will now choose exactly $k$ random vertices ($k \geq 3$) from her cake and cut a piece, the shape of which is the convex polygon defined by those vertices. Compute the expected area of this piece of cake.

Each test case will begin with a line with two space-separated integers $n$ and $k$ ($3 \leq k \leq n \leq 2{,}500$), where $n$ is the number of vertices of the cake, and $k$ is the number of vertices of the piece that Alice cuts. Each of the next $n$ lines will contain two space-separated real numbers $x$ and $y$ ($-10.0 \leq x, y \leq 10.0$), where $(x, y)$ is a vertex of the cake. The vertices will be listed in clockwise order. No three vertices will be collinear. All real numbers have at most 6 digits after the decimal point.

Output a single real number, which is the expected area of the piece of cake that Alice cuts out. Your answer will be accepted if it is within an absolute error of $10^{-6}$.