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: Runtime Error)

h, w, k = readline.split.map(&:to_i)
MOD = 1000000007

oy = (1...w).map{|i|(w-i)*i}.reduce(&:+) % MOD
oy = h*h * oy % MOD

ox = (1...h).map{|i| (h-i)*i}.reduce(&:+) % MOD
ox = w*w * ox % MOD

#a = p (0...h).flat_map{|x| (0...w).map{|y| [x,y]}}.combination(2).to_a
#p a.map{|((x1, y1),(x2,y2))| (x1-x2).abs}.reduce(&:+)
#p a.map{|((x1, y1),(x2,y2))| (y1-y2).abs}.reduce(&:+)

n = w * h
FACT = Array.new(n+1){1}
INV = Array.new(n+1){1}
FINV = Array.new(n+1){1}

(2..n).each{|i|
  FACT[i] = FACT[i-1] * i % MOD
  INV[i] = MOD - INV[MOD % i] * (MOD / i) % MOD
  FINV[i] = FINV[i-1] * INV[i] % MOD
}

def ncr n, r
  return 0 if n < r or n < 0 or r < 0
  return FACT[n] * FINV[r] * FINV[n - r] % MOD
end

al = oy + ox
als = ncr(h*w-2, k-2) * al % MOD
p als