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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Collections.Generic;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
using Debug = System.Diagnostics.Debug;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;

static class P
{
    static void Main()
    {
        var nabc = Console.ReadLine().Split().Select(int.Parse).ToArray();
        var n = nabc[0];
        ModInt a = new ModInt(nabc[1]) / (nabc[1] + nabc[2]);
        ModInt b = new ModInt(nabc[2]) / (nabc[1] + nabc[2]);
        ModInt c = new ModInt(nabc[3]) / 100;

        ModInt curPowA = 1;
        ModInt curPowB = 1;

        ModInt winA = Power(a, n);
        ModInt winB = Power(b, n);

        ModInt inv = 1 / Factorial(n - 1);

        ModInt res = 0;
        for (int i = n; i < n * 2; i++)
        {
            //引き分けを考えずに、a/bがi回勝って上がる確率→
            //winのn-1回, loseのi-n回 の並び替え 合計i-1Cn-1
            //
            res += i * ((curPowA * winB + curPowB * winA) * Factorial(i - 1) * inv / Factorial(i - n));
            curPowA *= a;
            curPowB *= b;
        }
        //100/c
        res /= (1 - c);
        Console.WriteLine(res);

        //10回上がったら終わりのシークエンスについて、どれだけ引き分けが入り込む余地があるか
        //逆数
        /*Random rng = new Random();
        double res = 0;
        for (int i = 0; i < 10000; i++)
        {
            int count = 0;
            int totalCount = 0;
            while (count < 10)
            {
                if (rng.NextDouble() < 0.3333) count++;
                totalCount++;
            }
            res += totalCount;
        }
        Console.WriteLine(res / 10000);
        */
    }

    static List<ModInt> factorialMemo = new List<ModInt>() { 1 };
    static ModInt Factorial(int x)
    {
        for (int i = factorialMemo.Count; i <= x; i++) factorialMemo.Add(factorialMemo.Last() * i);
        return factorialMemo[x];
    }
    static ModInt Power(ModInt n, long m)
    {
        ModInt pow = n;
        ModInt res = 1;
        while (m > 0)
        {
            if ((m & 1) == 1) res *= pow;
            pow *= pow;
            m >>= 1;
        }
        return res;
    }
}



struct ModInt
{
    const int MOD = 1000000007;
    long Data;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public ModInt(long data) { if ((Data = data % MOD) < 0) Data += MOD; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator long(ModInt modInt) => modInt.Data;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator ModInt(long val) => new ModInt(val);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt a, ModInt b)
    {
        long res = a.Data + b.Data;
        return new ModInt() { Data = res >= MOD ? res - MOD : res };
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt a, long b) => a.Data + b;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt a, ModInt b)
    {
        long res = a.Data - b.Data;
        return new ModInt() { Data = res < 0 ? res + MOD : res };
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt a, long b) => a.Data - b;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator *(ModInt a, ModInt b) => new ModInt() { Data = a.Data * b.Data % MOD };
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator /(ModInt a, ModInt b) => a.Data * GetInverse(b);

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public override string ToString() => Data.ToString();

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static long GetInverse(long a)
    {
        long div;
        long p = MOD;
        long x1 = 1, y1 = 0, x2 = 0, y2 = 1;
        while (true)
        {
            if (p == 1) return x2;
            div = a / p;
            x1 -= x2 * div;
            y1 -= y2 * div;
            a %= p;
            if (a == 1) return x1;
            div = p / a;
            x2 -= x1 * div;
            y2 -= y1 * div;
            p %= a;
        }
    }
}