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.

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 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 |