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: Accepted)

integer(16) N,M,K
integer(16) waruno
integer(16) ans,tmp,i,j,com
read*,N,M,K
waruno=10**9+7
com=combination(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"(i0)",ans

contains

function combination(n,k,l)
implicit none
integer(16),intent(in)::n,k,l
integer(16) combination,i
integer(16),allocatable,dimension(:)::Kaijo

!まず最初に階乗について計算します
allocate(Kaijo(0:N))
Kaijo(0)=1
do i=1,N
  Kaijo(i)=mod(Kaijo(i-1)*i,l)
end do

!そこから階乗を使ってコンビネーションを計算します
!invはmod 10**9+7の下での逆数です
combination=mod(Kaijo(N)*inv(Kaijo(K),l)*inv(Kaijo(N-K),l),l)

end function combination

function inv(K,L)
integer(16),intent(in)::k,l
integer(16) l2,y
integer(16) inv
!フェルマーの小定理から-1乗とは10**9+7-2乗だよねということで
!10**9+7-2乗を計算します

!計算に使うのは繰り返し二乗法
inv=1
l2=l-2
y=k
do
  if(and(l2,1)==1)inv=mod(inv*y,l)
  y=mod(y*y,l)
  l2=rshift(l2,1)
  if(l2==0)exit
end do
end function inv
end