Problem p03039 (C++)

Buggy submission (status: Runtime Error):
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <iomanip>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <vector>

using namespace std;
using lint = long long;
constexpr int MOD = 1000000007, INF = 1010101010;
constexpr lint LINF = 1LL << 60;

template <class T> ostream &operator<<(ostream &os, const vector<T> &vec) {
  for (const auto &e : vec)
    os << e << (&e == &vec.back() ? "\n" : " ");
  return os;
}

#ifdef _DEBUG
template <class T> void dump(const char *str, T &&h) {
  cerr << str << " = " << h << "\n";
};
template <class Head, class... Tail>
void dump(const char *str, Head &&h, Tail &&...t) {
  while (*str != ',')
    cerr << *str++;
  cerr << " = " << h << "\n";
  dump(str + (*(str + 1) == ' ' ? 2 : 1), t...);
}
#define DMP(...) dump(#__VA_ARGS__, __VA_ARGS__)
#else
#define DMP(...) ((void)0)
#endif

template <int Modulo = MOD> struct Mint {

  lint val;
  constexpr Mint(lint v = 0) noexcept : val(v % Modulo) {
    if (val < 0)
      val += Modulo;
  }

  constexpr Mint &operator+=(const Mint &r) noexcept {
    val += r.val;
    if (val >= Modulo)
      val -= Modulo;
    return *this;
  }
  constexpr Mint &operator-=(const Mint &r) noexcept {
    val -= r.val;
    if (val < 0)
      val += Modulo;
    return *this;
  }
  constexpr Mint &operator*=(const Mint &r) noexcept {
    val = val * r.val % Modulo;
    return *this;
  }
  constexpr Mint &operator/=(const Mint &r) noexcept {
    lint a = r.val, b = Modulo, u = 1, v = 0;
    while (b) {
      lint t = a / b;
      a -= t * b;
      swap(a, b);
      u -= t * v;
      swap(u, v);
    }
    val = val * u % Modulo;
    if (val < 0)
      val += Modulo;
    return *this;
  }

  constexpr Mint operator+(const Mint &r) const noexcept {
    return Mint(*this) += r;
  }
  constexpr Mint operator-(const Mint &r) const noexcept {
    return Mint(*this) -= r;
  }
  constexpr Mint operator*(const Mint &r) const noexcept {
    return Mint(*this) *= r;
  }
  constexpr Mint operator/(const Mint &r) const noexcept {
    return Mint(*this) /= r;
  }

  constexpr int getmod() { return Modulo; }
  constexpr Mint operator-() const noexcept { return val ? Modulo - val : 0; }

  constexpr bool operator==(const Mint &r) const noexcept {
    return val == r.val;
  }
  constexpr bool operator!=(const Mint &r) const noexcept {
    return val != r.val;
  }

  friend ostream &operator<<(ostream &os, const Mint<Modulo> &x) noexcept {
    return os << x.val;
  }
  friend istream &operator>>(istream &is, Mint<Modulo> &x) noexcept {
    lint tmp;
    is >> tmp;
    x = Mint<Modulo>(tmp);
    return is;
  }
  friend constexpr Mint<Modulo> modpow(const Mint<Modulo> &a, lint n) noexcept {
    Mint res(1), tmp = a;
    while (n > 0) {
      if (n & 1)
        res *= tmp;
      tmp *= tmp;
      n >>= 1;
    }
    return res;
  }
};

using mint = Mint<>;

class Factorial {
private:
  vector<mint> fac, inv, finv;

  void build(int N) {
    fac[0] = fac[1] = 1;
    inv[1] = 1;
    finv[0] = finv[1] = 1;

    for (int i = 2; i < N; i++) {
      fac[i] = fac[i - 1] * i;
      inv[i] = -inv[MOD % i] * (MOD / i);
      finv[i] = finv[i - 1] * inv[i];
    }
  }

public:
  Factorial(int N = 110000) : fac(N + 1), inv(N + 1), finv(N + 1) {
    build(N + 1);
  }

  mint operator()(int n, int k) {
    if (n < k || k < 0)
      return 0LL;
    return fac[n] * (finv[k] * finv[n - k]);
  }

  mint Pmod(int n, int k) {
    if (n < k || k < 0)
      return 0LL;
    return fac[n] * finv[n - k];
  }
};

int main() {

  cin.tie(nullptr);
  ios::sync_with_stdio(false);

  int N, M, K;
  cin >> N >> M >> K;

  mint ans = 0;

  ans += mint(1) * M * M * (N - 1) * N * (N + 1) / 6;
  ans += mint(1) * N * N * (M - 1) * M * (M + 1) / 6;
  ans *= Factorial()(N * M - 2, K - 2);

  cout << ans << "\n";

  return 0;
}

Fixed submission:
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <iomanip>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <vector>

using namespace std;
using lint = long long;
constexpr int MOD = 1000000007, INF = 1010101010;
constexpr lint LINF = 1LL << 60;

template <class T> ostream &operator<<(ostream &os, const vector<T> &vec) {
  for (const auto &e : vec)
    os << e << (&e == &vec.back() ? "\n" : " ");
  return os;
}

#ifdef _DEBUG
template <class T> void dump(const char *str, T &&h) {
  cerr << str << " = " << h << "\n";
};
template <class Head, class... Tail>
void dump(const char *str, Head &&h, Tail &&...t) {
  while (*str != ',')
    cerr << *str++;
  cerr << " = " << h << "\n";
  dump(str + (*(str + 1) == ' ' ? 2 : 1), t...);
}
#define DMP(...) dump(#__VA_ARGS__, __VA_ARGS__)
#else
#define DMP(...) ((void)0)
#endif

template <int Modulo = MOD> struct Mint {

  lint val;
  constexpr Mint(lint v = 0) noexcept : val(v % Modulo) {
    if (val < 0)
      val += Modulo;
  }

  constexpr Mint &operator+=(const Mint &r) noexcept {
    val += r.val;
    if (val >= Modulo)
      val -= Modulo;
    return *this;
  }
  constexpr Mint &operator-=(const Mint &r) noexcept {
    val -= r.val;
    if (val < 0)
      val += Modulo;
    return *this;
  }
  constexpr Mint &operator*=(const Mint &r) noexcept {
    val = val * r.val % Modulo;
    return *this;
  }
  constexpr Mint &operator/=(const Mint &r) noexcept {
    lint a = r.val, b = Modulo, u = 1, v = 0;
    while (b) {
      lint t = a / b;
      a -= t * b;
      swap(a, b);
      u -= t * v;
      swap(u, v);
    }
    val = val * u % Modulo;
    if (val < 0)
      val += Modulo;
    return *this;
  }

  constexpr Mint operator+(const Mint &r) const noexcept {
    return Mint(*this) += r;
  }
  constexpr Mint operator-(const Mint &r) const noexcept {
    return Mint(*this) -= r;
  }
  constexpr Mint operator*(const Mint &r) const noexcept {
    return Mint(*this) *= r;
  }
  constexpr Mint operator/(const Mint &r) const noexcept {
    return Mint(*this) /= r;
  }

  constexpr int getmod() { return Modulo; }
  constexpr Mint operator-() const noexcept { return val ? Modulo - val : 0; }

  constexpr bool operator==(const Mint &r) const noexcept {
    return val == r.val;
  }
  constexpr bool operator!=(const Mint &r) const noexcept {
    return val != r.val;
  }

  friend ostream &operator<<(ostream &os, const Mint<Modulo> &x) noexcept {
    return os << x.val;
  }
  friend istream &operator>>(istream &is, Mint<Modulo> &x) noexcept {
    lint tmp;
    is >> tmp;
    x = Mint<Modulo>(tmp);
    return is;
  }
  friend constexpr Mint<Modulo> modpow(const Mint<Modulo> &a, lint n) noexcept {
    Mint res(1), tmp = a;
    while (n > 0) {
      if (n & 1)
        res *= tmp;
      tmp *= tmp;
      n >>= 1;
    }
    return res;
  }
};

using mint = Mint<>;

class Factorial {
private:
  vector<mint> fac, inv, finv;

  void build(int N) {
    fac[0] = fac[1] = 1;
    inv[1] = 1;
    finv[0] = finv[1] = 1;

    for (int i = 2; i < N; i++) {
      fac[i] = fac[i - 1] * i;
      inv[i] = -inv[MOD % i] * (MOD / i);
      finv[i] = finv[i - 1] * inv[i];
    }
  }

public:
  Factorial(int N = 110000) : fac(N + 1), inv(N + 1), finv(N + 1) {
    build(N + 1);
  }

  mint operator()(int n, int k) {
    if (n < k || k < 0)
      return 0LL;
    return fac[n] * (finv[k] * finv[n - k]);
  }

  mint Pmod(int n, int k) {
    if (n < k || k < 0)
      return 0LL;
    return fac[n] * finv[n - k];
  }
};

int main() {

  cin.tie(nullptr);
  ios::sync_with_stdio(false);

  int N, M, K;
  cin >> N >> M >> K;

  mint ans = 0;

  ans += mint(1) * M * M * (N - 1) * N * (N + 1) / 6;
  ans += mint(1) * N * N * (M - 1) * M * (M + 1) / 6;
  ans *= Factorial(N * M)(N * M - 2, K - 2);

  cout << ans << "\n";

  return 0;
}