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: Accepted)

N,M,K=gets.split.map(&:to_i)
MOD=1000000007

def xsum
  sum=0
  (N-1).times do |i|
    i+=1
    sum+=(i*(N-i)*M*M) % MOD
  end
  return sum
end

def ysum
  sum=0
  (M-1).times do |i|
    i+=1
    sum+=(i*(M-i)*N*N) % MOD
  end
  return sum
end

def comb(n, k)
  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
    return ret
  end
  k = k < n-k ? k : n-k
  return (n-k+1..n).inject(1){|p, q| p*q % MOD} * mpow((1 .. k).inject(1){|p, q| p*q % MOD}, MOD - 2)
end
puts ((xsum+ysum)*comb(N*M-2,K-2)) % MOD