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 Kotlin (1.0.0) (judge verdict: Accepted)

fun main(args : Array<String>) {
    val (N, M, K) = readLine()!!.split(" ").map { it.toInt() }

    val MOD = 1000000007L
    val allConbination = nCk(N*M -2, K -2, MOD.toInt()) % MOD
    val NN = N * N % MOD
    val MM = M * M % MOD

    var costX = 0L
    var costY = 0L
    (1..M).forEach { m ->
        costX += allConbination * NN * m * (M-m) % MOD
    }
    (1..N).forEach { n ->
        costX += allConbination * MM * n * (N-n) % MOD
    }
    println((costX + costY) % MOD)
}


fun pow(a: Long, b: Long, M: Int): Long {
    var b = b
    var ret: Long = 1
    var tmp = a
    while (b > 0) {
        if (b and 1 == 1L) ret = ret * tmp % M
        tmp = tmp * tmp % M
        b = b shr 1
    }
    return ret
}

//return nCk mod M (M must be prime number) O(min(k,n-k)*logM)
fun nCk(n: Int, k: Int, M: Int): Int {
    var ret: Long = 1
    val min = Math.min(k, n - k)
    for (i in 1..min) {
        ret = ret * pow(i.toLong(), (M - 2).toLong(), M) % M
    }
    for (i in n - min + 1..n) {
        ret = ret * i % M
    }
    return ret.toInt()
}