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 Python (3.8.2) (judge verdict: Accepted)

#!/usr/bin/env python3
def main():
    N, M, K = map(int, input().split())
    m = 10**9+7
    # 方針: ２つの駒の距離だけ考えれば良い、それ以外K-2個の駒の位置が(N*M-2)C(K-2)通りある
    # マンハッタン距離はx方向とy方向を独立に扱える
    ans = 0
    for d in range(1,M):
        ans += (d*(M-d)%m) * pow(N,2,m) % m
        ans %= m

    for d in range(1,N):
        ans += (d*(N-d)%m) * pow(M,2,m) % m
        ans %= m

    # 答えに(NM-2)C(K-2)かける
    # K回しか処理しないから間に合う
    for r in range(K-2):
        ans *= ((N*M%m-2)%m-r)%m
        ans %= m
        ans *= pow(r+1,m-2,m)
        ans %= m
    print(ans)

if __name__ == "__main__":
    main()