Problem Description

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.

Input Format

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 Format

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 $10^{-6}$.

4 3
0 0
1 1
2 1
1 0

0.50000000

5 5
0 4
4 2
4 1
3 -1
-2 4

12.50000000

5 3
-1.20 2.80
3.30 2.40
3.10 -0.80
2.00 -4.60
-4.40 -0.50

12.43300000