Problem:
Score : 500 points

Problem Statement

We have a grid of squares with N rows and M columns. Let (i, j) denote the square at the i-th row from the top and j-th column from the left. We will choose K of the squares and put a piece on each of them.

If we place the K pieces on squares (x_1, y_1), (x_2, y_2), ..., and (x_K, y_K), the cost of this arrangement is computed as:

\sum_{i=1}^{K-1} \sum_{j=i+1}^K (|x_i - x_j| + |y_i - y_j|)

Find the sum of the costs of all possible arrangements of the pieces. Since this value can be tremendous, print it modulo 10^9+7.

We consider two arrangements of the pieces different if and only if there is a square that contains a piece in one of the arrangements but not in the other.

Constraints

2 \leq N \times M \leq 2 \times 10^5

2 \leq K \leq N \times M

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M K

Output

Print the sum of the costs of all possible arrangements of the pieces, modulo 10^9+7.

Sample Input 1

2 2 2

Sample Output 1

8

There are six possible arrangements of the pieces, as follows:

((1,1),(1,2)), with the cost |1-1|+|1-2| = 1

((1,1),(2,1)), with the cost |1-2|+|1-1| = 1

((1,1),(2,2)), with the cost |1-2|+|1-2| = 2

((1,2),(2,1)), with the cost |1-2|+|2-1| = 2

((1,2),(2,2)), with the cost |1-2|+|2-2| = 1

((2,1),(2,2)), with the cost |2-2|+|1-2| = 1

The sum of these costs is 8.

Sample Input 2

4 5 4

Sample Output 2

87210

Sample Input 3

100 100 5000

Sample Output 3

817260251

Be sure to print the sum modulo 10^9+7.

Solution in Scala (2.11.7) (judge verdict: Accepted)

import java.io._
import java.util.StringTokenizer
import scala.util.control.Breaks.{breakable,break}
import scala.collection.mutable.{Map, Stack, Queue, PriorityQueue}

object Main extends App {
  val in = new InputReader(System.in)
  val pw = new PrintWriter(System.out)

  val N, M, K = in.next().toLong

  val MOD = 1e9.toLong + 7

  var ans = 0L

  val combN = (N*M - 2).toInt

  val mc = new ModComb(combN, MOD)

  def calc(x: Long, y: Long, dist: Long): Long = {
    var res = y*y * (x-dist)
    res %= MOD
    res *= mc.comb(combN, K.toInt - 2)
    res %= MOD
    res *= dist
    res %= MOD

    res
  }

  for (dist <- 1 to (N-1).toInt) {
    ans += calc(N, M, dist)
    ans %= MOD
  }

  for (dist <- 1 to (M-1).toInt) {
    ans += calc(M, N, dist)
    ans %= MOD
  }

  println(ans)
}

class ModComb(maxNum: Int, mod: Long) {

  private def repeatSquares(x: Long, n: Long): Long = {
    if(n == 0)
      1
    else if(n % 2 == 0)
      repeatSquares(x*x % mod, n/2)
    else
      (x * repeatSquares(x*x % mod, n/2)) % mod
  }

  private val fact = new Array[Long](maxNum + 1)

  fact(0) = 1
  for (i <- 1 to maxNum)
    fact(i) = (i * fact(i-1)) % mod

  private val factInv = new Array[Long](maxNum + 1)

  factInv(maxNum) = repeatSquares(fact(maxNum), mod-2)
  for (i <- (1 to maxNum).reverse)
    factInv(i-1) = (i * factInv(i)) % mod


  // nCk
  def comb(n: Int, k: Int): Long = {
    val numer = fact(n)
    val denom = (factInv(n-k) * factInv(k)) % mod

    (numer * denom) % mod
  }

}

class InputReader(stream: InputStream) {
  private[this] val reader = new BufferedReader(new InputStreamReader(stream))
  private[this] var tokenizer: StringTokenizer = new StringTokenizer(reader.readLine())
  def next(): String = {
    while (!tokenizer.hasMoreTokens()) {
      tokenizer = new StringTokenizer(reader.readLine())
    }
    tokenizer.nextToken()
  }
}

object Sub {
  def print2DimArray[A](mat: Array[Array[A]]): Unit = {
    mat.foreach(lin => println(lin.deep))
  }
}