output_description: Print the sum of the costs of all possible arrangements of the pieces, modulo
  10^9+7.
  
  * * *
input_description: Input is given from Standard Input in the following format:
  
      
      
      N M K
attempt: N, M, K = list(map(int, input().split()))
  
  MOD = int(1e9 + 7)
  NM = N * M
  
  
  def su(n):
      return n * (n + 1) // 2
  
  
  class comb:
      F = [1, 1]
      Fi = [1, 1]
      I = [0, 1]
  
      def __init__(self, num, mod):
          self.MOD = mod
          for i in range(2, num + 1):
              self.F.append((self.F[-1] * i) % mod)
              self.I.append(mod - self.I[mod % i] * (mod // i) % mod)
              self.Fi.append(self.Fi[-1] * self.I[i] % mod)
  
      def com(self, n, k):
          if n < k:
              return 0
          if n < 0 or k < 0:
              return 0
          return self.F[n] * (self.Fi[k] * self.Fi[n - k] % self.MOD) % self.MOD
  
  
  co = comb(NM, MOD)
  times = co.com(NM - 2, K - 2)
  Ans = 0
  for i in range(N):
      for j in range(M):
          k = (su(i) + su(N - i - 1)) * M % MOD
          k = k + (su(j) + su(M - j - 1)) * N % MOD
          k = k * times // 2 % MOD
          Ans += k % MOD
          Ans = Ans % MOD
  
  print(Ans)
  
problem_description: 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.
samples: [{"input": "2 2 2", "output": "8\n    \n\nThere are six possible arrangements of the pieces, as follows:\n\n  * ((1,1),(1,2)), with the cost |1-1|+|1-2| = 1\n  * ((1,1),(2,1)), with the cost |1-2|+|1-1| = 1\n  * ((1,1),(2,2)), with the cost |1-2|+|1-2| = 2\n  * ((1,2),(2,1)), with the cost |1-2|+|2-1| = 2\n  * ((1,2),(2,2)), with the cost |1-2|+|2-2| = 1\n  * ((2,1),(2,2)), with the cost |2-2|+|1-2| = 1\n\nThe sum of these costs is 8.\n\n* * *"}, {"input": "4 5 4", "output": "87210\n    \n\n* * *"}, {"input": "100 100 5000", "output": "817260251\n    \n\nBe sure to print the sum modulo 10^9+7."}]