Problem p02808 (C++)

Buggy submission (status: Time Limit Exceeded):
#include <bits/stdc++.h>
#define LLI long long int
#define FOR(v, a, b) for (LLI v = (a); v < (b); ++v)
#define FORE(v, a, b) for (LLI v = (a); v <= (b); ++v)
#define REP(v, n) FOR(v, 0, n)
#define REPE(v, n) FORE(v, 0, n)
#define REV(v, a, b) for (LLI v = (a); v >= (b); --v)
#define ALL(x) (x).begin(), (x).end()
#define RALL(x) (x).rbegin(), (x).rend()
#define ITR(it, c) for (auto it = (c).begin(); it != (c).end(); ++it)
#define RITR(it, c) for (auto it = (c).rbegin(); it != (c).rend(); ++it)
#define EXIST(c, x) ((c).find(x) != (c).end())
#define fst first
#define snd second
#define popcount __builtin_popcount
#define UNIQ(v) (v).erase(unique(ALL(v)), (v).end())
#define bit(i) (1LL << (i))

#ifdef DEBUG
#include <misc/C++/Debug.cpp>
#else
#define dump(...) ((void)0)
#endif

#define gcd __gcd

using namespace std;
template <class T> constexpr T lcm(T m, T n) { return m / gcd(m, n) * n; }

template <typename I> void join(ostream &ost, I s, I t, string d = " ") {
  for (auto i = s; i != t; ++i) {
    if (i != s)
      ost << d;
    ost << *i;
  }
  ost << endl;
}
template <typename T> istream &operator>>(istream &is, vector<T> &v) {
  for (auto &a : v)
    is >> a;
  return is;
}
template <typename T, typename U> bool chmin(T &a, const U &b) {
  return (a > b ? a = b, true : false);
}
template <typename T, typename U> bool chmax(T &a, const U &b) {
  return (a < b ? a = b, true : false);
}
template <typename T, size_t N, typename U>
void fill_array(T (&a)[N], const U &v) {
  fill((U *)a, (U *)(a + N), v);
}

struct Init {
  Init() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(12);
    cerr << fixed << setprecision(12);
  }
} init;

template <std::uint32_t M> class ModInt {
public:
  std::uint64_t val;
  ModInt() : val(0) {}
  ModInt(std::int64_t n) {
    if (n >= M)
      val = n % M;
    else if (n < 0)
      val = n % M + M;
    else
      val = n;
  }

  inline constexpr ModInt operator+(const ModInt &a) const {
    return ModInt((val + a.val) % M);
  }
  inline constexpr ModInt operator-(const ModInt &a) const {
    return ModInt((val - a.val + M) % M);
  }
  inline constexpr ModInt operator*(const ModInt &a) const {
    return ModInt((val * a.val) % M);
  }
  inline constexpr ModInt operator/(const ModInt &a) const {
    return ModInt((val * a.inv().val) % M);
  }

  inline constexpr ModInt &operator=(const ModInt &a) {
    val = a.val;
    return *this;
  }
  inline constexpr ModInt &operator+=(const ModInt &a) {
    if ((val += a.val) >= M)
      val -= M;
    return *this;
  }
  inline constexpr ModInt &operator-=(const ModInt &a) {
    if (val < a.val)
      val += M;
    val -= a.val;
    return *this;
  }
  inline constexpr ModInt &operator*=(const ModInt &a) {
    (val *= a.val) %= M;
    return *this;
  }
  inline constexpr ModInt &operator/=(const ModInt &a) {
    (val *= a.inv().val) %= M;
    return *this;
  }

  inline constexpr bool operator==(const ModInt &a) const {
    return val == a.val;
  }
  inline constexpr bool operator!=(const ModInt &a) const {
    return val != a.val;
  }

  inline constexpr ModInt &operator++() {
    *this += 1;
    return *this;
  }
  inline constexpr ModInt &operator--() {
    *this -= 1;
    return *this;
  }

  inline constexpr ModInt operator++(int) {
    ModInt t = *this;
    *this += 1;
    return t;
  }
  inline constexpr ModInt operator--(int) {
    ModInt t = *this;
    *this -= 1;
    return t;
  }

  inline constexpr static ModInt frac(std::int64_t a, std::int64_t b) {
    return ModInt(a) / ModInt(b);
  }

  inline constexpr static ModInt power(std::int64_t n, std::int64_t p) {
    if (p < 0)
      return power(n, -p).inv();

    ModInt ret = 1, e = n;
    for (; p; e *= e, p >>= 1)
      if (p & 1)
        ret *= e;
    return ret;
  }

  inline constexpr ModInt power(std::int64_t p) const { return power(val, p); }

  inline constexpr ModInt inv() const {
    std::int64_t a = val, b = M, u = 1, v = 0;

    while (b) {
      std::int64_t t = a / b;
      a -= t * b;
      std::swap(a, b);
      u -= t * v;
      std::swap(u, v);
    }
    u %= M;
    if (u < 0)
      u += M;

    return u;
  }
};

template <std::uint32_t M> ModInt<M> operator-(const ModInt<M> &a) {
  return M - a.val;
}

template <std::uint32_t M>
ModInt<M> operator+(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) + b.val);
}
template <std::uint32_t M>
ModInt<M> operator-(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) - b.val);
}
template <std::uint32_t M>
ModInt<M> operator*(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) * b.val);
}
template <std::uint32_t M>
ModInt<M> operator/(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) / b.val);
}

template <std::uint32_t M>
std::istream &operator>>(std::istream &is, ModInt<M> &a) {
  is >> a.val;
  return is;
}
template <std::uint32_t M>
std::ostream &operator<<(std::ostream &os, const ModInt<M> &a) {
  os << a.val;
  return os;
}

/**
 * @attention 使用前にinit関数を呼び出す
 */
template <typename T> class Combinatorics {
public:
  static std::vector<T> facto;
  static std::vector<T> ifacto;

  static void init(int N) {
    facto.assign(N + 1, 1);
    ifacto.assign(N + 1, 1);

    for (int i = 1; i <= N; ++i) {
      facto[i] = facto[i - 1] * i;
    }

    ifacto[N] = facto[N].inv();

    for (int i = N - 1; i >= 0; --i) {
      ifacto[i] = ifacto[i + 1] * (i + 1);
    }
  }

  static T f(int64_t i) {
    assert(i < facto.size());
    return facto[i];
  }

  static T finv(int64_t i) {
    assert(i < ifacto.size());
    return ifacto[i];
  }

  static T P(int64_t n, int64_t k);
  static T C(int64_t n, int64_t k);
  static T H(int64_t n, int64_t k);
  static T stirling_number(int64_t n, int64_t k);
  static T bell_number(int64_t n, int64_t k);
  static std::vector<T> bernoulli_number(int64_t n);
  static T catalan_number(int64_t n);
};

template <typename T> std::vector<T> Combinatorics<T>::facto = std::vector<T>();
template <typename T>
std::vector<T> Combinatorics<T>::ifacto = std::vector<T>();

template <typename T> T Combinatorics<T>::P(int64_t n, int64_t k) {
  if (n < k or n < 0 or k < 0)
    return 0;
  return f(n) * finv(n - k);
}

template <typename T> T Combinatorics<T>::C(int64_t n, int64_t k) {
  if (n < k or n < 0 or k < 0)
    return 0;
  return P(n, k) * finv(k);
}

template <typename T> T Combinatorics<T>::H(int64_t n, int64_t k) {
  if (n == 0 and k == 0)
    return 1;
  return C(n + k - 1, k);
}

using mint = ModInt<1000000007>;
using C = Combinatorics<mint>;

int main() {
  C::init(100000);

  int N, K;
  while (cin >> N >> K) {
    vector<int> a(K);
    cin >> a;
    dump(a);

    vector<vector<mint>> dp(K + 1, vector<mint>(N + 1, 0));

    dp[0][0] = 1;

    REP(i, K) {
      REPE(j, N) {
        REPE(x, N) {
          if (j + x <= N)
            dp[i + 1][j + x] += dp[i][j] * C::C(N - x, a[i] - x) / C::f(x);
        }
      }
    }

    mint ans = dp[K][N] * C::f(N);

    cout << ans << endl;
  }

  return 0;
}


Fixed submission:
#include <bits/stdc++.h>
#define LLI long long int
#define FOR(v, a, b) for (LLI v = (a); v < (b); ++v)
#define FORE(v, a, b) for (LLI v = (a); v <= (b); ++v)
#define REP(v, n) FOR(v, 0, n)
#define REPE(v, n) FORE(v, 0, n)
#define REV(v, a, b) for (LLI v = (a); v >= (b); --v)
#define ALL(x) (x).begin(), (x).end()
#define RALL(x) (x).rbegin(), (x).rend()
#define ITR(it, c) for (auto it = (c).begin(); it != (c).end(); ++it)
#define RITR(it, c) for (auto it = (c).rbegin(); it != (c).rend(); ++it)
#define EXIST(c, x) ((c).find(x) != (c).end())
#define fst first
#define snd second
#define popcount __builtin_popcount
#define UNIQ(v) (v).erase(unique(ALL(v)), (v).end())
#define bit(i) (1LL << (i))

#ifdef DEBUG
#include <misc/C++/Debug.cpp>
#else
#define dump(...) ((void)0)
#endif

#define gcd __gcd

using namespace std;
template <class T> constexpr T lcm(T m, T n) { return m / gcd(m, n) * n; }

template <typename I> void join(ostream &ost, I s, I t, string d = " ") {
  for (auto i = s; i != t; ++i) {
    if (i != s)
      ost << d;
    ost << *i;
  }
  ost << endl;
}
template <typename T> istream &operator>>(istream &is, vector<T> &v) {
  for (auto &a : v)
    is >> a;
  return is;
}
template <typename T, typename U> bool chmin(T &a, const U &b) {
  return (a > b ? a = b, true : false);
}
template <typename T, typename U> bool chmax(T &a, const U &b) {
  return (a < b ? a = b, true : false);
}
template <typename T, size_t N, typename U>
void fill_array(T (&a)[N], const U &v) {
  fill((U *)a, (U *)(a + N), v);
}

struct Init {
  Init() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(12);
    cerr << fixed << setprecision(12);
  }
} init;

template <std::uint32_t M> class ModInt {
public:
  std::uint64_t val;
  ModInt() : val(0) {}
  ModInt(std::int64_t n) {
    if (n >= M)
      val = n % M;
    else if (n < 0)
      val = n % M + M;
    else
      val = n;
  }

  inline constexpr ModInt operator+(const ModInt &a) const {
    return ModInt((val + a.val) % M);
  }
  inline constexpr ModInt operator-(const ModInt &a) const {
    return ModInt((val - a.val + M) % M);
  }
  inline constexpr ModInt operator*(const ModInt &a) const {
    return ModInt((val * a.val) % M);
  }
  inline constexpr ModInt operator/(const ModInt &a) const {
    return ModInt((val * a.inv().val) % M);
  }

  inline constexpr ModInt &operator=(const ModInt &a) {
    val = a.val;
    return *this;
  }
  inline constexpr ModInt &operator+=(const ModInt &a) {
    if ((val += a.val) >= M)
      val -= M;
    return *this;
  }
  inline constexpr ModInt &operator-=(const ModInt &a) {
    if (val < a.val)
      val += M;
    val -= a.val;
    return *this;
  }
  inline constexpr ModInt &operator*=(const ModInt &a) {
    (val *= a.val) %= M;
    return *this;
  }
  inline constexpr ModInt &operator/=(const ModInt &a) {
    (val *= a.inv().val) %= M;
    return *this;
  }

  inline constexpr bool operator==(const ModInt &a) const {
    return val == a.val;
  }
  inline constexpr bool operator!=(const ModInt &a) const {
    return val != a.val;
  }

  inline constexpr ModInt &operator++() {
    *this += 1;
    return *this;
  }
  inline constexpr ModInt &operator--() {
    *this -= 1;
    return *this;
  }

  inline constexpr ModInt operator++(int) {
    ModInt t = *this;
    *this += 1;
    return t;
  }
  inline constexpr ModInt operator--(int) {
    ModInt t = *this;
    *this -= 1;
    return t;
  }

  inline constexpr static ModInt frac(std::int64_t a, std::int64_t b) {
    return ModInt(a) / ModInt(b);
  }

  inline constexpr static ModInt power(std::int64_t n, std::int64_t p) {
    if (p < 0)
      return power(n, -p).inv();

    ModInt ret = 1, e = n;
    for (; p; e *= e, p >>= 1)
      if (p & 1)
        ret *= e;
    return ret;
  }

  inline constexpr ModInt power(std::int64_t p) const { return power(val, p); }

  inline constexpr ModInt inv() const {
    std::int64_t a = val, b = M, u = 1, v = 0;

    while (b) {
      std::int64_t t = a / b;
      a -= t * b;
      std::swap(a, b);
      u -= t * v;
      std::swap(u, v);
    }
    u %= M;
    if (u < 0)
      u += M;

    return u;
  }
};

template <std::uint32_t M> ModInt<M> operator-(const ModInt<M> &a) {
  return M - a.val;
}

template <std::uint32_t M>
ModInt<M> operator+(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) + b.val);
}
template <std::uint32_t M>
ModInt<M> operator-(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) - b.val);
}
template <std::uint32_t M>
ModInt<M> operator*(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) * b.val);
}
template <std::uint32_t M>
ModInt<M> operator/(std::int64_t a, const ModInt<M> &b) {
  return ModInt<M>(ModInt<M>(a) / b.val);
}

template <std::uint32_t M>
std::istream &operator>>(std::istream &is, ModInt<M> &a) {
  is >> a.val;
  return is;
}
template <std::uint32_t M>
std::ostream &operator<<(std::ostream &os, const ModInt<M> &a) {
  os << a.val;
  return os;
}

/**
 * @attention 使用前にinit関数を呼び出す
 */
template <typename T> class Combinatorics {
public:
  static std::vector<T> facto;
  static std::vector<T> ifacto;

  static void init(int N) {
    facto.assign(N + 1, 1);
    ifacto.assign(N + 1, 1);

    for (int i = 1; i <= N; ++i) {
      facto[i] = facto[i - 1] * i;
    }

    ifacto[N] = facto[N].inv();

    for (int i = N - 1; i >= 0; --i) {
      ifacto[i] = ifacto[i + 1] * (i + 1);
    }
  }

  static T f(int64_t i) {
    assert(i < facto.size());
    return facto[i];
  }

  static T finv(int64_t i) {
    assert(i < ifacto.size());
    return ifacto[i];
  }

  static T P(int64_t n, int64_t k);
  static T C(int64_t n, int64_t k);
  static T H(int64_t n, int64_t k);
  static T stirling_number(int64_t n, int64_t k);
  static T bell_number(int64_t n, int64_t k);
  static std::vector<T> bernoulli_number(int64_t n);
  static T catalan_number(int64_t n);
};

template <typename T> std::vector<T> Combinatorics<T>::facto = std::vector<T>();
template <typename T>
std::vector<T> Combinatorics<T>::ifacto = std::vector<T>();

template <typename T> T Combinatorics<T>::P(int64_t n, int64_t k) {
  if (n < k or n < 0 or k < 0)
    return 0;
  return f(n) * finv(n - k);
}

template <typename T> T Combinatorics<T>::C(int64_t n, int64_t k) {
  if (n < k or n < 0 or k < 0)
    return 0;
  return P(n, k) * finv(k);
}

template <typename T> T Combinatorics<T>::H(int64_t n, int64_t k) {
  if (n == 0 and k == 0)
    return 1;
  return C(n + k - 1, k);
}

using mint = ModInt<1000000007>;
using C = Combinatorics<mint>;

int main() {
  C::init(100000);

  int N, K;
  while (cin >> N >> K) {
    vector<int> a(K);
    cin >> a;
    dump(a);

    vector<vector<mint>> dp(K + 1, vector<mint>(N + 1, 0));

    dp[0][0] = 1;

    REP(i, K) {
      REPE(j, N) {
        REPE(x, N) {
          if (j + x > N)
            break;
          dp[i + 1][j + x] += dp[i][j] * C::C(N - x, a[i] - x) * C::finv(x);
        }
      }
    }

    mint ans = dp[K][N] * C::f(N);

    cout << ans << endl;
  }

  return 0;
}