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 Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

main=do
  [n,m,k]<-map read.words<$>getLine
  print $ solve n m k

solve :: Integer->Integer->Integer->Integer
solve n m k = go n m +% go m n
  where go w h = h *% h *% comb (w*h-2) (k-2) *% foldl1 (+%) [b-a|a<-[1..w],b<-[a+1..w]]

comb n r= factm n /% factm r /% factm (n-r)

mo = 1000000007

infixl 7 *%
a *% b = a * b `mod` mo

infixl 7 /%
a /% b = a *% inv b

infixl 6 +%
a +% b = (a + b) `mod` mo

infixl 6 -%
a -% b = (a-b+mo) `mod` mo

factm = foldl (*%) 1 . enumFromTo 1

inv x = u where (u,_,_) = euc x mo

euc = go 1 0 0 1
  where
  go u v u' v' x 0 = (u,v,x)
  go u v u' v' x y = go u' v' (u - q * u') (v - q * v') y r
    where (q,r) = quotRem x y