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)

def fact(n)
  @memo ||= []
  @memo[n] ||= (n==0) ? 1 : (n * fact(n-1)) % MOD;
end

def modpow(x,n)
  ret = 1
  while n > 0
    ret = (ret * x) % MOD if n&1 == 1
    x = (x*x) % MOD
    n >>= 1
  end
  return ret
end

def combin(n,k)
  return (((fact(n) * modpow(fact(n-k),MOD-2)) % MOD) * modpow(fact(k),MOD-2)) % MOD
end

MOD = 10**9 + 7
n,m,k = gets.split.map(&:to_i)

(n*m).times do |i|
  fact(i)
end

x = [n,m].max
csum = [0]*(x+1)
1.upto(x) do |i|
  csum[i] = (csum[i-1] + i) % MOD
end

sum = 0
n.times do |i|
  m.times do |j|
    sum += (csum[i] * m) % MOD
    sum += (csum[j] * n) % MOD
    sum %= MOD
  end
end

mul = combin(n*m-2,k-2)
p (sum*mul) % MOD