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2019-03-02 18:00 UTC

NAIPC 2019

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2019-03-02 23:00 UTC
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Problem A
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 ($k{\ge }3$) 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 $k$ ($3\! \le \! k\! \le \! n\! \le \! 2\, 500$), 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 $y$ ($-10.0{\le }x,y{\le }10.0$), 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 $10^{-6}$.

Sample Input 1 Sample Output 1
4 3
0 0
1 1
2 1
1 0
0.50000000
Sample Input 2 Sample Output 2
5 5
0 4
4 2
4 1
3 -1
-2 4
12.50000000
Sample Input 3 Sample Output 3
5 3
-1.20 2.80
3.30 2.40
3.10 -0.80
2.00 -4.60
-4.40 -0.50
12.43300000