OI Problems

## #60v5d7. Piece of Cake

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$3\leq k\leq n\leq 2500$

## 样例输入输出

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


## 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

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

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