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.chomp.split(" ").map(&:to_i);

CD = 10 ** 9 + 7

s = 0
(0...M).each{|j|
  (0...N).each{|i|
    if j == 0 then
      s += (i + j) * (N - i) * (M - j)
    else
      s += (i + j) * (N - i) * (M - j) * (i == 0 ? 1 : 2)
    end
  }
}

a = N * M - 2
b = K - 2
c = 1
b = a - b if a < 2 * b
if b != 0 then
  bb = (1..b).to_a
  c = 1
  (a-b+1..a).each{|i|
    c *= i
    while (bb.size > 0) && (c % bb[0] == 0) do
      c /= bb[0]
      bb.shift
    end
  }
end

puts ((c % CD) * s) % CD