Problem:
Score : 500 points

Problem Statement

Takahashi and Aoki will play a game. They will repeatedly play it until one of them have N wins in total.

When they play the game once, Takahashi wins with probability A %, Aoki wins with probability B %, and the game ends in a draw (that is, nobody wins) with probability C %.
Find the expected number of games that will be played, and print it as follows.

We can represent the expected value as P/Q with coprime integers P and Q.
Print the integer R between 0 and 10^9+6 (inclusive) such that R \times Q \equiv P\pmod {10^9+7}.
(Such an integer R always uniquely exists under the constraints of this problem.)

Constraints

1 \leq N \leq 100000

0 \leq A,B,C \leq 100

1 \leq A+B

A+B+C=100

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N A B C

Output

Print the expected number of games that will be played, in the manner specified in the statement.

Sample Input 1

1 25 25 50

Sample Output 1

2

Since N=1, they will repeat the game until one of them wins.
The expected number of games played is 2.

Sample Input 2

4 50 50 0

Sample Output 2

312500008

C may be 0.

Sample Input 3

1 100 0 0

Sample Output 3

1

B may also be 0.

Sample Input 4

100000 31 41 28

Sample Output 4

104136146

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

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

typedef long long ll;
typedef unsigned long long ull;
typedef pair<int,int> pii;
typedef pair<ll, ll> pll;
typedef pair<ull, ull> pullull;
typedef pair<ll, int> plli;
typedef pair<int, pii> pipii;
typedef vector<vector<int> > mati;
typedef vector<vector<double> > matd;
typedef vector<ll> vll;
typedef vector<vector<ll>> vvll;
typedef vector<vector<vector<ll>>> vvvll;
typedef vector<bool> vb;
typedef vector<vector<bool>> vvb;
typedef vector<vector<vector<bool>>> vvvb;
typedef vector<pll> vpll;

#define FOR(i,x,y) for(ll i=(ll)x; i<(ll)y; ++i)
#define REP(i,y) FOR(i, 0, y)
#define RFOR(i,x,y) for(ll i=(ll)x; i>=(ll)y; --i)
#define RREP(i,x) RFOR(i, x, 0)
#define ALL(a) a.begin(), a.end()
#define pb push_back

inline void IN(void){
  return;
}

template <typename First, typename... Rest>
void IN(First& first, Rest&... rest){
  cin >> first;
  IN(rest...);
  return;
}

inline void OUT(void){
  cout << "\n";
  return;
}

template <typename First, typename... Rest>
void OUT(First first, Rest... rest){
  cout << first << " ";
  OUT(rest...);
  return;
}

template <typename T>
void vec_print(vector<T> VEC){
  REP(i, VEC.size()){
    cout << VEC[i] << " ";
  }
  cout << "\n";
};

template <typename T>
void mat_print(vector<vector<T> > MAT){
  REP(i, MAT.size()){
    REP(j, MAT[i].size()){
      cout << MAT[i][j] << " ";
    }
    cout << "\n";
  }
};

template <typename CLASS1, typename CLASS2>
class HOGE{
  public:
    CLASS1 key;
    CLASS2 value;
    HOGE(void){
      return;
    };
    HOGE(CLASS1 key, CLASS2 value){
      this->key = key;
      this->value = value;
    };
    ~HOGE(void){
      return;
    };

    void print(void){
      cout << "key : " << key << ", value : " << value << "\n";
      return;
    };
    
    bool operator==(const HOGE &obj){
      return (this->value == obj.value);
    };
    bool operator<(const HOGE &obj){
      return (this->value < obj.value);
    };
    bool operator>(const HOGE &obj){
      return (this->value > obj.value);
    };
};

template <typename CLASS1, typename CLASS2>
bool operator==(const HOGE<CLASS1, CLASS2> &hoge1, const HOGE<CLASS1, CLASS2> &hoge2){
  return hoge1.value == hoge2.value;
};

template <typename CLASS1, typename CLASS2>
bool operator<(const HOGE<CLASS1, CLASS2> &hoge1, const HOGE<CLASS1, CLASS2> &hoge2){
  return hoge1.value < hoge2.value;
};

template <typename CLASS1, typename CLASS2>
bool operator>(const HOGE<CLASS1, CLASS2> &hoge1, const HOGE<CLASS1, CLASS2> &hoge2){
  return hoge1.value > hoge2.value;
};

constexpr int INF = (1<<30);
constexpr ll INFLL = 1LL<<62;
constexpr long double EPS = 1e-12;
constexpr ll MOD = (ll)((1E+9)+7);

class Modulo{
  private:
    ll MAX = 200000;
    vll fac_v, finv_v, inv_v;
  public:

    void Modulo_setting(ll MAX_){
      MAX = MAX_;

      fac_v.resize(MAX);
      finv_v.resize(MAX);
      inv_v.resize(MAX);

      fac_v[0] = fac_v[1] = 1;
      finv_v[0] = finv_v[1] = 1;
      inv_v[1] = 1;
      for(int i=2; i<=MAX; ++i){
        //i!
        fac_v[i] = (fac_v[i-1]*i)%MOD;
        //i^{-1}
        inv_v[i] = (MOD-(inv_v[MOD%i]*(MOD/i))%MOD)%MOD;
        //(i!)^{-1}
        finv_v[i] = (finv_v[i-1]*inv_v[i])%MOD;
      }
    };

    Modulo(ll MAX_){
      Modulo_setting(MAX_);
      return;
    };

    Modulo(void){
      Modulo_setting(MOD);
      return;
    };

    ~Modulo(void){
      return;
    };

    ll inv(ll n){
      if(n<0 || n>MAX){
        fprintf(stderr, "n range error\n");
        exit(0);
      }
      return inv_v[n];
    };

    ll fac(ll n){
      if(n<0 || n>MAX){
        fprintf(stderr, "n range error\n");
        exit(0);
      }
      return fac_v[n];
    };

    ll inv_fac(ll n){
      if(n<0 || n>MAX){
        fprintf(stderr, "n range error\n");
        exit(0);
      }
      return finv_v[n];
    };

    ll Comb(ll n, ll k){
      if(n<k) return 0;
      if(n<0 || k<0) return 0;
      return (fac_v[n]*((finv_v[k]*finv_v[n-k])%MOD))%MOD;
    };

};

// TakがN回勝って終わる期待値
// N A^(N-1)*A + NC1 (N+1) A^(N-1)B^1 * A + (N+1)C2 (N+2) A^(N-1)B^2 * A + ... + (2N-2)C(N-1) (2N-1) A^(N-1)B^(N-1) * A
// = A^N (N + (N+1) NC1 B + (N+2) (N+1)C2 B^2 + ... + (2N-1) (2N-2)C(N-1) B^(N-1))
// = A^N (N + (N+1)C2 2B + (N+2)C3 3B^2 + ... + (2N-1)CN NB^(N-1)
//

ll N, A, B, C;
Modulo que(200000);
vll dpA, dpB;

ll calc(void){
  ll ans = 0;
  REP(i,N){
    ans += (((N+i)* que.Comb(N-1+i, i))%MOD) * ((dpA[N] * dpB[i])%MOD + (dpB[N] * dpA[i])%MOD);
    ans %= MOD;
  }
  return ans;
}

ll calc_inv(ll x){
  if(x==1) return (ll)1;
  return (MOD-(calc_inv(MOD%x)*(MOD/x))%MOD)%MOD;
}

int main(){
  cin.tie(0); // cut the cin and cout (default, std::flush is performed after std::cin)
  ios::sync_with_stdio(false); // cut the iostream and stdio (DON'T endl; BUT "\n";)

  IN(N,A,B,C);

  ll inv_100 = que.inv(100);
  A = (A*inv_100)%MOD;
  B = (B*inv_100)%MOD;
  ll inv = calc_inv((A+B)%MOD);

   A = (A*inv)%MOD;
   B = (B*inv)%MOD;

  dpA.resize(N+1, 1);
  dpB.resize(N+1, 1);
  REP(i,N){
    dpA[i+1] = (dpA[i]*A)%MOD;
    dpB[i+1] = (dpB[i]*B)%MOD;
  }

  ll ans = calc();
  ans *= inv;
  ans %= MOD;


  OUT(ans);

  return 0;
}