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 PyPy3 (2.4.0) (judge verdict: Accepted)

facs = []
M = 1000000007

def modulo(n):
    global facs
    facs = [1]*(n+1)
    last = 1
    for i in range(1,n+1):
        last = facs[i] = mulmod(last,i,M)
        
def mulmod(x,y,p):
    return x*y % p
def divmod(x,y,p):
    return mulmod(x,powmod(y,p-2,p),p)

def ncr(n,r):
    if n<r:
        return 0
    if n==r:
        return 1
    res = facs[n]
    res = divmod(res,facs[r],M)
    res = divmod(res,facs[n-r],M)
    return res

def powmod(x,y,p):
    if y==0:
        return 1
    elif y==1:
        return x % p
    elif (y%2)==0:
        return powmod(x,y//2,p)**2 % p
    else:
        return powmod(x,y//2,p)**2 * x % p
def resolve():
    n,m,k = map(int,input().split())
    modulo(2*10**5+5)
    kumi=ncr(n*m-2,k-2)
    ans = 0
    for i in range(1,n):
        ans = ans + ((i*(n-i)*m*m)%M)
    for i in range(1,m):
        ans = ans + ((i*(m-i)*n*n)%M)
    print(ans*kumi%M)

if __name__ == "__main__":
    resolve()