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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

N, M, K = gets.split.map(&:to_i)
MOD = 1E9.to_i + 7

def fact(n)
  (1 .. n).inject(1){|r, x| r*x % MOD}
end

def mpow(x, n)
  ret = 1
  while n > 0
    if n.even?
      x = x * x % MOD
      n >>= 1
    else
      ret = ret * x % MOD
      n -= 1
    end
  end
  ret
end

def comb(n, k)
  fact(n) * mpow(fact(k), MOD - 2) * mpow(fact(n - k), MOD - 2)
end

def f(x, y)
  (1 .. x - 1).inject(0){|s, d| (s + comb(x*y - 2, K - 2) * d * (x - d)*y*y) % MOD}
end

puts (f(N, M) + f(M, N)) % MOD