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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <string>
#include <algorithm>
#include <cmath>
#include <map>
#include <vector>
#include <stack>
#include <queue>
#include <tuple>
#include <cassert>
#include <functional>
#include <set>
#include <deque>
using namespace std;
typedef long long ll;
int dx[4] = {0,0,1,-1};
int dy[4] = {1,-1,0,0};
const int INF = 1<<29;
#define LNF __LONG_LONG_MAX__;
const ll MOD = 1000000007;
const int MAX = 200000;

long long fac[MAX], finv[MAX], inv[MAX];

// テーブルを作る前処理
void COMinit() {
    fac[0] = fac[1] = 1;
    finv[0] = finv[1] = 1;
    inv[1] = 1;
    for (int i = 2; i < MAX; i++){
        fac[i] = fac[i - 1] * i % MOD;
        inv[i] = MOD - inv[MOD%i] * (MOD / i) % MOD;
        finv[i] = finv[i - 1] * inv[i] % MOD;
    }
}

// 二項係数計算
long long COM(int n, int k){
    if (n < k) return 0;
    if (n < 0 || k < 0) return 0;
    return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
}
// 法mでのaの逆元を計算
long long modinv(long long a, long long m) {
    long long b = m, u = 1, v = 0;
    while (b) {
        long long t = a / b;
        a -= t * b; swap(a, b);
        u -= t * v; swap(u, v);
    }
    u %= m;
    if (u < 0) u += m;
    return u;
}
// a^n mod を計算する
long long modpow(long long a, long long n, long long mod) {
    long long res = 1;
    while (n > 0) {
        if (n & 1) res = res * a % mod;
        a = a * a % mod;
        n >>= 1;
    }
    return res;
}
//最大公約数
int gcd(int x, int y) { return (x % y)? gcd(y, x % y): y; }
//最小公倍数
int lcm(int x, int y) { return x / gcd(x, y) * y; }
// nが大きく,kが10^7程度の時の二項係数
long long com[200005];
void combination(long long n,long long k){
    com[0] = 1;
    for(int i=1;i<=k;i++){
        com[i] = (com[i-1]*(n-i+1)%MOD)*modinv(i,MOD)%MOD;
    }
}

int main(){
    int N; cin >> N;
    vector<pair<int,int> > x(N);
    vector<int> y(N);
    for(int i=0;i<N;i++){
        int a,b; cin >> a >> b;
        x[i] = make_pair(a,b);
        y[i] = i;
    }
    double average=0.0;
    int cnt=0;
    do{
        cnt++;
        cerr << "\n" << y[0] << " ";
        double dist=0.0;
        for(int i=1;i<N;i++){
            cerr << y[i] << " ";
            int sx=x[y[i-1]].first,sy=x[y[i-1]].second;
            int gx=x[y[i]].first,gy=x[y[i]].second;
            dist += sqrt(1.0*(sx-gx)*(sx-gx)+1.0*(sy-gy)*(sy-gy));
        }
        average += dist;
    }while(next_permutation(y.begin(),y.end()));
    printf("%.8lf\n",average/cnt);
    return 0;
}