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 Fortran (gfortran v4.8.4) (judge verdict: Runtime Error)

integer(16) N,M,K
integer(16) waruno
integer(16) ans,tmp,i,j,com
read*,N,M,K
waruno=10**9+7
com=comb(N*M-2,K-2,waruno)
ans=0
do i=0,M-1
  do j=0,N-1
    tmp=mod((i+j)*(M-i)*(N-j)*com*merge(1,2,i*j==0),waruno)
    ans=mod(ans+tmp,waruno)
  end do
end do
print*,ans

contains
pure function comb(n,k,waruno)
integer(16),intent(in)::n,k,waruno
integer(16) i,j
integer(16),allocatable,dimension(:,:)::Table
integer(16) comb
allocate(Table(0:n,0:n))
Table=0
Table(:,0)=1
do i=1,N
  do j=1,i
    Table(i,j)=mod(Table(i-1,j-1)+Table(i-1,j),waruno)
  end do
end do
comb=Table(n,k)
end function comb
end