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 (3.4.3) (judge verdict: Accepted)

N,M,K=map(int,input().split())
mod=10**9+7
NM=N*M
kaijo=[0]*(NM+1)
gyaku=[0]*(NM+1)
kaijo[0]=1
for i in range(1,NM+1):
    kaijo[i]=kaijo[i-1]*i%mod
gyaku[NM]=pow(kaijo[NM],mod-2,mod)
for i in range(NM,0,-1):
    gyaku[i-1]=gyaku[i]*i%mod
C=kaijo[NM-2]*gyaku[K-2]*gyaku[NM-K]%mod
ans=0
for k in range(1,M):
    x=k*(M-k)*C*N*N%mod
    ans=(ans+x)%mod
for k in range(1,N):
    x=k*(N-k)*C*M*M%mod
    ans=(ans+x)%mod
print(ans)