Problem:
Score : 900 points

Problem Statement

There is a rectangular sheet of paper with height H+1 and width W+1. We introduce an xy-coordinate system so that the four corners of the sheet are (0, 0), (W + 1, 0), (0, H + 1) and (W + 1, H + 1).

This sheet can be cut along the lines x = 1,2,...,W and the lines y = 1,2,...,H. Consider a sequence of operations of length K where we choose K of these H + W lines and cut the sheet along those lines one by one in some order.

Let the score of a cut be the number of pieces of paper that exist just after the cut. The score of a sequence of operations is the sum of the scores of all of the K cuts.

Find the sum of the scores of all possible sequences of operations of length K. Since this value can be extremely large, print the number modulo 10^9 + 7.

Constraints

1 \leq H,W \leq 10^7

1 \leq K \leq H + W

H, W and K are integers.

Input

Input is given from Standard Input in the following format:

H W K

Output

Print the sum of the scores, modulo 10^9 + 7.

Sample Input 1

2 1 2

Sample Output 1

34

Let x_1, y_1 and y_2 denote the cuts along the lines x = 1, y = 1 and y = 2, respectively. The six possible sequences of operations and the score of each of them are as follows:

y_1, y_2: 2 + 3 = 5

y_2, y_1: 2 + 3 = 5

y_1, x_1: 2 + 4 = 6

y_2, x_1: 2 + 4 = 6

x_1, y_1: 2 + 4 = 6

x_1, y_2: 2 + 4 = 6

The sum of these is 34.

Sample Input 2

30 40 50

Sample Output 2

616365902

Be sure to print the sum modulo 10^9 + 7.

Solution in C++14 (GCC 5.4.1) (judge verdict: Runtime Error)

#include <bits/stdc++.h>
#include <functional>
using namespace std;

void recursive_comb(int *indexes,long long int s,long long int rest, std::function<void(int *)> f) {
  if (rest == 0) {
    f(indexes);
  } else {
    if (s < 0) return;
    recursive_comb(indexes, s - 1, rest, f);
    indexes[rest - 1] = s;
    recursive_comb(indexes, s - 1, rest - 1, f);
  }
}

// nCkの組み合わせに対して処理を実行する
void foreach_comb(long long int n, long long int k, std::function<void(int *)> f) {
  int indexes[k];
  recursive_comb(indexes, n - 1, k, f);
}

// nPnの順列に対して処理を実行する
void foreach_permutation(long long int n, std::function<void(int *)> f) {
  int indexes[n];
  for (int i = 0; i < n; i++) indexes[i] = i;
  do {
    f(indexes);
  } while (std::next_permutation(indexes, indexes + n));
}

// nPkの順列に対して処理を実行する
void foreach_permutation(long long int n, long long int k, std::function<void(int *)> f) {
  foreach_comb(n, k, [&](int *c_indexes) {
    foreach_permutation(k, [&](int *p_indexes) {
      int indexes[k];
      for (long long int i = 0; i < k; i++) {
        indexes[i] = c_indexes[p_indexes[i]];
      }
      f(indexes);
    });
  });
}

int main(){
 	 long long int h,w,kkk,sum=0;
 	 cin >> h >> w >> kkk;
 	 foreach_permutation(h+w, kkk, [&](int *indexes) {
        long long int kk[kkk],hcut=0,wcut=0;
        for (int i=0;i<kkk;i++){
        	if (indexes[i]>=w){
            	hcut++;
         	}
          	else {
          		wcut++;
          	}
        	kk[i]=(hcut+1)*(wcut+1);
       		sum += kk[i];
       		sum %= 1000000007;
     	 }
     });
	cout << sum << endl;

  
}