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 Scala (2.11.7) (judge verdict: Accepted)

import java.io.PrintWriter

object Scanner {
  private val buf = new Array[Byte](1024); private var ptr = 0; private var len = 0
  @inline private def isPrintableChar(c: Int): Boolean = 33 <= c && c <= 126
  @inline private def hasNextByte(): Boolean =
    if (ptr >= len) { ptr = 0; len = System.in.read(buf); len > 0 } else { true }
  @inline private def hasNext(): Boolean = {
    while (hasNextByte() && !isPrintableChar(buf(ptr))) ptr += 1
    hasNextByte()
  }
  @inline private def readByte(): Byte =
    if (hasNextByte()) { val res = buf(ptr); ptr += 1; res } else { -1 }
  def next(): String = {
    if(!hasNext()) ???
    val sb = new StringBuilder; var b = readByte()
    while (isPrintableChar(b)) { sb.append(b.toChar); b = readByte() }
    sb.toString
  }
  def nextInt(): Int = {
    val n = nextLong()
    if (n < Int.MinValue || Int.MaxValue < n) ???
    n.toInt
  }
  def nextLong(): Long = {
    if(!hasNext()) ???
    var minus = false; var b = readByte()
    if (b == '-') { minus = true; b = readByte() }
    def go (b: Byte, n: Long = 0): Long =
      if ('0' <= b && b <= '9') { go(readByte(), n * 10 + b - '0') }
      else if (minus) { -n } else { n }
    go(b)
  }
  def nextDouble(): Double = next.toDouble
}

object OutPut {
  def print(f:PrintWriter => Unit): Unit = {
    val pw: PrintWriter = new java.io.PrintWriter(System.out)
    //pw.println("")
    f(pw)
    pw.flush()
  }
}
object MyDataFormat {
  def double(x: Double, precision: Int): String = {
    var num = x
    val sb = new StringBuilder
    if (num < 0) {
      sb.append("-")
      num = -num
    }
    num += Math.pow(10, -precision) / 2
    sb.append(num.toLong)
    sb.append(".")
    num -= num.toLong
    (0 until precision).foreach(i => {
      num *= 10
      sb.append(num.toInt)
      num -= num.toInt
    })
    sb.toString
  }

  def array2(arr: scala.collection.mutable.ArrayBuffer[scala.collection.mutable.ArrayBuffer[Int]], x: Int, y: Int): Option[Int] = {
    try { Some(arr(x)(y)) } catch {
      case _: Exception => None
    }
  }
}

object Loop {
  @inline def rep(n: Int, f: => Unit): Unit = { var c = 0; while (c < n) { f; c += 1 } }
  @inline def rep(n: Int, f: Int => Unit): Unit = { var c = 0; while (c < n) { f(c); c += 1 } }
  @inline def !(n: Int): Int = (1 + n) * n / 2
}

object Search {
  def find(data: Array[Int], key: Int): Option[Int] = {
    val res = java.util.Arrays.binarySearch(data, key)
    if (res >= 0) Some(res) else None
  }
}

object WarshallFloyd {
  def run(n: Int, vectorMap: Array[Array[Int]]): Array[Array[Int]] = {
    Loop.rep(n, k => {
      Loop.rep(n, i => {
        Loop.rep(n, j => {
          val a = vectorMap(i)(j)

          val b = vectorMap(i)(k)
          val c = vectorMap(k)(j)
          val distance = Math.min(a, b + c)
          vectorMap(i)(j) = distance
        })
      })
    })
    vectorMap
  }
}

case class CumulativeSum(arr: Array[Int]) {
  val n = arr.length
  val sumArray: Array[Int] = Array.fill[Int](n + 1)(0)
  Loop.rep(n, i => sumArray(i + 1) = sumArray(i) + arr(i))

  //TODO 半開区間に変更
  //座標指定 上限下限共に含む
  def countByCoordinate(start: Int, end: Int): Int =
    countBySubscript(start - 1, end - 1)

  //添字指定 上限下限共に含む
  def countBySubscript(start: Int, end: Int): Int = {
    if (start <= end) sumArray(end + 1) - sumArray(start)
    else sumArray(start + 1) - sumArray(end)
  }
}

case class UnionFind(n: Int) {
  //添字が固有ID、値が親IDを表す
  val arr: Array[Int] = Array.tabulate[Int](n)(identity)
  def findParent(v: Int): Int = {
    val parent = arr(v)
    if (v == parent) parent else findParent(parent)
  }
  def registerParent(child: Int, parent: Int): Unit = {
    val oldParent = arr(child)
    arr(oldParent) = parent
  }
}

object PrimeNumbers {
  def isPrimeNumber(n: Int): Boolean = ???
  def createUntil(n: Int): Array[Int] = {
    val arr: Array[Int] = Array.empty
    //i = 2からスタートしたいので全体をずらす
    Loop.rep(n - 2, i => {
      //自身より小さい全ての素数で割り切れなければ、素数
      if (!arr.exists(v => i % v == 0)) {
        //append
      }
    })
    arr
  }
}



object Main extends App {
  val n = Scanner.nextInt()
  val data = Array.fill[Int](n, 2)(Scanner.nextInt())

  var sum: Double = 0
  Loop.rep(n, x => {
    Loop.rep(n, y => {
      if (y > x) {
        val a = data(x)
        val b = data(y)
        val dx: Double = a(0) - b(0)
        val dy: Double = a(1) - b(1)
        sum += Math.sqrt(dx * dx + dy * dy)
      }
    })
  })

  println(sum / n * 2)
}