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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.Arrays;
import java.util.Scanner;

public class Main {

    private static int MOD = 1_000_000_007;

    public static void main(String[] args) {
        final Scanner scanner = new Scanner(System.in);
        solve(scanner);
    }

    // int N = Integer.parseInt(scanner.nextLine());
    // int N = scanner.nextInt(), M = scanner.nextInt();
    // scanner.nextLine();
    // int[] n = Arrays.stream(scanner.nextLine().split(" ")).mapToInt(Integer::parseInt).toArray();
    // long[] n = Arrays.stream(scanner.nextLine().split(" ")).mapToLong(Long::parseLong).toArray();
    static void solve(Scanner scanner) {
        int[] n = Arrays.stream(scanner.nextLine().split(" ")).mapToInt(Integer::parseInt).toArray();
        int N = n[0], M = n[1], K = n[2];

        Combination combination = new Combination(200_000, MOD);
        long x = 0, y = 0;
        for (int d = 1; d < N; d++) {
            x = (x + d * (N - d) * M * M) % MOD;
        }
        for (int d = 1; d < M; d++) {
            y = (y + d * (M - d) * N * N) % MOD;
        }
        long abs = (combination.kFromN(K - 2, N * M - 2) * (x + y)) % MOD;
        System.out.println(abs);
    }

}

class Combination {

    private int MOD;
    private int MAX;

    private final long[] fac, facInv, inv;

    public Combination(int max, int mod) {
        this.MAX = max + 1;
        this.MOD = mod;
        fac = new long[MAX];
        facInv = new long[MAX];
        inv = new long[MAX];

        fac[0] = fac[1] = 1;
        facInv[0] = facInv[1] = 1;
        inv[1] = 1;
        for (int i = 2; i < MAX; i++) {
            fac[i] = fac[i - 1] * i % MOD;
            inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
            facInv[i] = facInv[i - 1] * inv[i] % MOD;
        }
    }

    public long kFromN(int k, int n) {
        if (n < k) return 0;
        if (n < 0 || k < 0) return 0;
        return fac[n] * (facInv[k] * facInv[n - k] % MOD) % MOD;
    }
}