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: Accepted)

// includes {{{
#include<iostream>
#include<iomanip>
#include<algorithm>
#include<vector>
#include<stack>
#include<queue>
#include<map>
#include<set>
#include<tuple>
#include<cmath>
#include<random>
#include<cassert>
// #include<deque>
// #include<multiset>
// #include<bitset>
// #include<cstring>
// #include<bits/stdc++.h>
// }}}
using namespace std;
using ll = long long;

// WARN : use H with larger N
/// --- Modulo Factorial {{{ ///
template < int N, int mod = (int) 1e9 + 7 >
struct Factorial {
  constexpr ll extgcd(ll a, ll b, ll &x, ll &y) {
    ll d = 0;
    return b == 0 ? (x = a < 0 ? -1 : 1, y = 0, a < 0 ? -a : a)
      : (d = extgcd(b, a % b, y, x), y -= a / b * x, d);
  }
  constexpr ll modinv(ll a) {
    ll x = 0, y = 0;
    extgcd(a, mod, x, y);
    return (x + mod) % mod;
  }
  int arr[N + 1], inv[N + 1];
  ll operator[](int i) const { return arr[i]; }
    Factorial()
    : arr(), inv() {
      arr[0] = 1;
      for(int i = 1; i <= N; i++) {
        arr[i] = (ll) i * arr[i - 1] % mod;
      }
      inv[N] = modinv(arr[N]);
      for(int i = N - 1; i >= 0; i--) {
        inv[i] = (ll)(i + 1) * inv[i + 1] % mod;
      }
    }
  ll C(int n, int r) const {
    if(n < 0 || r < 0 || n < r) return 0;
    return (ll) arr[n] * inv[r] % mod * inv[n - r] % mod;
  }
  ll H(int n, int r) const { return C(n + r - 1, r); }
};
/// }}}--- ///

const int N = 1e7;
Factorial < 2 * N > fact;

/// --- ModInt Library {{{ ///
#include <ostream>
template < ll mod = (ll) 1e9 + 7 >
struct ModInt {
  // math {{{
  static inline ll extgcd(ll a, ll b, ll &x, ll &y) {
    ll d;
    return b == 0 ? (x = a < 0 ? -1 : 1, y = 0, a < 0 ? -a : a)
      : (d = extgcd(b, a % b, y, x), y -= a / b * x, d);
  }
  static inline ll modinv(ll a) {
    ll x = 0, y = 0;
    extgcd(a, mod, x, y);
    if(x < 0) x += mod;
    return x;
  }
  static inline ll modpow(ll a, ll b) {
    if(b < 0) b = -b, a = modinv(a);
    ll r = 1;
    a %= mod;
    while(b) {
      if(b & 1) r = r * a % mod;
      a = a * a % mod;
      b >>= 1;
    }
    return r;
  }
  // }}}

  ll val;
  constexpr ModInt() : val(0) {}
  constexpr ModInt(ll t) {
    val = t % mod;
    if(val < 0) val += mod;
  }

private:
  // strict constructor
  constexpr ModInt(ll t, int) : val(t) {}

public:
  template < class T >
    explicit operator T() {
      return T(val);
    }
  // ModInt <arithmetic-operator>[=] ModInt {{{
  ModInt operator+(ModInt const &rhs) const {
    ModInt tmp = *this;
    tmp += rhs;
    return tmp;
  }
  ModInt operator-(ModInt const &rhs) const {
    ModInt tmp = *this;
    tmp -= rhs;
    return tmp;
  }
  ModInt operator*(ModInt const &rhs) const {
    ModInt tmp = *this;
    tmp *= rhs;
    return tmp;
  }
  ModInt operator/(ModInt const &rhs) const {
    ModInt tmp = *this;
    tmp /= rhs;
    return tmp;
  }
  ModInt &operator+=(ModInt const &rhs) {
    val = val + rhs.val;
    if(val >= mod) val -= mod;
    return *this;
  }
  ModInt &operator-=(ModInt const &rhs) { return *this += -rhs; }
  ModInt &operator*=(ModInt const &rhs) {
    val = val * rhs.val % mod;
    return *this;
  }
  ModInt &operator/=(ModInt const &rhs) { return *this *= rhs.inv(); }
  // }}}
  // increment, decrement {{{
  ModInt operator++(int) {
    ModInt tmp = *this;
    val = val + 1;
    if(val >= mod) val = 0;
    return tmp;
  }
  ModInt operator--(int) {
    ModInt tmp = *this;
    val = val == 0 ? mod - 1 : val - 1;
    return tmp;
  }
  ModInt &operator++() {
    val = val + 1;
    if(val >= mod) val = 0;
    return *this;
  }
  ModInt &operator--() {
    val = val == 0 ? mod - 1 : val - 1;
    return *this;
  }
  // }}}
  ModInt operator-() const { return ModInt(val == 0 ? 0 : mod - val, 0); }
  // ModInt <arithmetic-operator>[=] T {{{
  template < typename T >
    ModInt operator+(T const &rhs) const {
      return ModInt(val + rhs % mod);
    }
  template < typename T >
    ModInt operator-(T const &rhs) const {
      return ModInt(mod + val - rhs % mod);
    }
  template < typename T >
    ModInt operator*(T const &rhs) const {
      return ModInt(val * (rhs % mod));
    }
  template < typename T >
    ModInt operator/(T const &rhs) const {
      return ModInt(val * modinv(rhs));
    }
  template < typename T >
    ModInt &operator+=(T const &rhs) {
      val = (mod + val + rhs % mod) % mod;
      return *this;
    }
  template < typename T >
    ModInt &operator-=(T const &rhs) {
      val = (mod + val - rhs % mod) % mod;
      return *this;
    }
  template < typename T >
    ModInt &operator*=(T const &rhs) {
      val = val * (mod + rhs % mod) % mod;
      return *this;
    }
  template < typename T >
    ModInt &operator/=(T const &rhs) {
      val = val * modinv(rhs) % mod;
      return *this;
    }
  // }}}
  ModInt inv() const { return ModInt(modinv(val), 0); }
  ModInt operator~() const { return inv(); }
  friend ostream &operator<<(ostream &os, ModInt const &mv) {
    os << mv.val;
    return os;
  }
  // equality operator {{{
  ModInt operator==(const ModInt &a) const { return val == a.val; }
  ModInt operator!=(const ModInt &a) const { return val != a.val; }
  ModInt operator==(const ll &a) const { return val == ModInt(a); }
  ModInt operator!=(const ll &a) const { return val != ModInt(a); }
  // }}}
  // T <arithmetic-operator> ModInt {{{
  friend constexpr ModInt operator+(ll a, ModInt const &mv) {
    return ModInt(a % mod + mv.val);
  }
  friend constexpr ModInt operator-(ll a, ModInt const &mv) {
    return ModInt(a % mod - mv.val);
  }
  friend constexpr ModInt operator*(ll a, ModInt const &mv) {
    return ModInt((mod + a % mod) * mv.val % mod, 0);
  }
  friend constexpr ModInt operator/(ll a, ModInt const &mv) {
    return ModInt((mod + a % mod) * modinv(mv.val) % mod, 0);
  }
  // }}}
  // power {{{
  ModInt operator^(ll x) const { return pow(*this, x); }
  ModInt &operator^=(ll x) {
    val = modpow(val, x);
    return *this;
  }
  friend ModInt pow(ModInt x, ll y) { return ModInt(modpow(x.val, y), 0); }
  // }}}
};
/// }}}--- ///

vector<ll> construct_modinvs(int n, ll mod = ll(1e9 + 7)) {
  vector<ll> res(n + 1);
  res[1] = 1;
  for(int i = 2; i <= n; i++) res[i] = mod - res[mod % i] * (mod / i) % mod;
  return res;
}

using modint = ModInt<>;

int main() {
  ios::sync_with_stdio(false), cin.tie(0);
  int h, w, k;
  cin >> h >> w >> k;
  // 全部のカットH+W個を並べて最初のK個を操作列として採用する，と考える
  auto invs = construct_modinvs(2*N);
  modint ans;
  for(int t = 1; t <= k; t++) {
    ans += (modint) 2 * invs[h + w] * (t - 1) * invs[h + w - 1] * (k + 1 - t) * h * w;
    ans += (modint) 1 * invs[h + w] * (k + 1 - t) * (h + w);
  }
  ans += k;
  // ansは期待値

  ans *= fact[h + w];
  // ansは期待値 * 個数
  // ただし同じものを (h + w - k)! 回数えている

  ans *= fact.inv[h + w - k];

  cout << ans << endl;
  return 0;
}