Problem:
Score : 300 points

Problem Statement

There are N towns in a coordinate plane. Town i is located at coordinates (x_i, y_i). The distance between Town i and Town j is \sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2}.

There are N! possible paths to visit all of these towns once. Let the length of a path be the distance covered when we start at the first town in the path, visit the second, third, \dots, towns, and arrive at the last town (assume that we travel in a straight line from a town to another). Compute the average length of these N! paths.

Constraints

2 \leq N \leq 8

-1000 \leq x_i \leq 1000

-1000 \leq y_i \leq 1000

\left(x_i, y_i\right) \neq \left(x_j, y_j\right) (if i \neq j)

(Added 21:12 JST) All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the average length of the paths.
Your output will be judges as correct when the absolute difference from the judge's output is at most 10^{-6}.

Sample Input 1

3
0 0
1 0
0 1

Sample Output 1

2.2761423749

There are six paths to visit the towns: 1 → 2 → 3, 1 → 3 → 2, 2 → 1 → 3, 2 → 3 → 1, 3 → 1 → 2, and 3 → 2 → 1.

The length of the path 1 → 2 → 3 is \sqrt{\left(0-1\right)^2+\left(0-0\right)^2} + \sqrt{\left(1-0\right)^2+\left(0-1\right)^2} = 1+\sqrt{2}.

By calculating the lengths of the other paths in this way, we see that the average length of all routes is:

\frac{\left(1+\sqrt{2}\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)}{6} = 2.276142...

Sample Input 2

2
-879 981
-866 890

Sample Output 2

91.9238815543

There are two paths to visit the towns: 1 → 2 and 2 → 1. These paths have the same length.

Sample Input 3

8
-406 10
512 859
494 362
-955 -475
128 553
-986 -885
763 77
449 310

Sample Output 3

7641.9817824387

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.PrintStream;
import java.util.Scanner;

public class Main {
	public static void main(String[] args) throws Exception{
		new Main(new Scanner(System.in), System.out, System.err).exec();
	}
	public final Scanner sc;
	public final PrintStream out, err;
	public Main(final Scanner sc, final PrintStream out, final PrintStream err) {
		this.sc = sc;this.out = out;this.err = err;
	}
	static private final long pair(int a, int b) {return (((long)a) << 32) + b;}
	static private final int upper(long v) {return (int)(v >> 32);}
	static private final int lower(long v) {return (int)(v & 0xFFFFFFFF);}

	//-----------------------------------------

	static private final long load(int end, int length) {
		return pair(end, length);
	}
	static private final int end(long load) {
		return upper(load);
	}
	static private final int length(long load) {
		return lower(load);
	}
	static private final long score(int count, int last) {
		return pair(count, last);
	}
	static private final int count(long score) {
		return upper(score);
	}
	static private final int last(long score) {
		return lower(score);
	}

	//------------------------------------------

	public void exec() throws Exception{
		final int N = sc.nextInt();
		final long[][] P = new long[N][2];
		final int x = 0;
		final int y = 1;

		for(int i=0;i<N;i++) {
			P[i][x] = sc.nextLong();
			P[i][y] = sc.nextLong();
		}

		final double[][] D = new double[N][N];
		for(int i=0;i<N;i++) {
			for(int j=0;j<N;j++) {
				if(i!=j) {
					D[i][j] = Math.sqrt(
							1.0*(
									(P[i][x]-P[j][x])*(P[i][x]-P[j][x]) +
									(P[i][y]-P[j][y])*(P[i][y]-P[j][y])
							));
				}
			}
		}
		boolean[] flags = new boolean[N];
		double sum = 0;
		int route[] = new int[N];
		cnt=0;
		for(int i=0;i<N;i++) {
			flags[i]=true;
			route[0]=i;
			sum += search(route, flags, D, 1, N, i, 0);
			flags[i]=false;
		}
//		err.println(sum);
//		err.println(N*(N+1)/2);
//		err.println(cnt);
		out.println(sum/cnt);
	}
	private int cnt = 0;

	private double search(int[] route, boolean[] flags, double[][] D, int depth, int N, int pos, double now) {
		if(depth == N-1) {
			cnt++;
			// 次が最終
			for(int i=0;i<N;i++) {
				if(!flags[i]) {
					route[depth]=i;
//					err.println(Arrays.toString(route));
					return now+D[pos][i];
				}
			}
		}
		double sum = 0;
		for(int i=0;i<N;i++) {
			if(!flags[i]) {
				flags[i]=true;
				route[depth]=i;
				sum+=search(route, flags, D, depth+1, N, i, now+D[pos][i]);
				flags[i]=false;
			}
		}
		return sum;
	}

}