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 Rust (1.15.1) (judge verdict: Accepted)

//! ----------------------------------------------
//! Framework <https://github.com/vain0x/procon>
//!
//! See the bottom of file for solution.
//! ----------------------------------------------

#![allow(unused_imports)]
#![allow(non_snake_case)]

use std::cell::RefCell;
use std::cmp::{max, min, Ordering};
use std::collections::*;
use std::fmt::{Debug, Display, Formatter, Write as FmtWrite};
use std::io::{stderr, stdin, BufRead, Write};
use std::mem::{replace, swap};
use std::ops::*;
use std::rc::Rc;

/// Print values to standard error if debug mode.
#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), stderr());
            writeln!(err, "\x1B[33m{}\x1B[0m = {:?}", e, $e).unwrap()
        })*
    };
}

/// Read from standard input and parse each word.
/// - `read!(T, U, ..)` parses a line as a tuple of words.
/// - `read![[T]]` parses a line as an array of words.
/// - `read![..; N]` parses `N` lines, using `read!(..)` repeatedly.
#[allow(unused_macros)]
macro_rules! read {
    ([$t:ty] ; $n:expr) =>
        ((0..$n).map(|_| read!([$t])).collect::<Vec<_>>());
    ($($t:ty),+ ; $n:expr) =>
        ((0..$n).map(|_| read!($($t),+)).collect::<Vec<_>>());
    ([$t:ty]) =>
        (rl().split_whitespace().map(|w| w.parse().unwrap()).collect::<Vec<$t>>());
    ($($t:ty),*) => {{
        let buf = rl();
        let mut w = buf.split_whitespace();
        ($(w.next().unwrap().parse::<$t>().unwrap()),*)
    }};
}

/// Read a line from standard input.
#[allow(dead_code)]
fn rl() -> String {
    let mut buf = String::new();
    stdin().read_line(&mut buf).unwrap();

    #[allow(deprecated)]
    buf.trim_right().to_owned()
}

// -----------------------------------------------
// Solution
// -----------------------------------------------

const P: i64 = 1_000_000_007;

pub fn pow(x: i64, n: i64) -> i64 {
    let (mut x, mut y, mut n) = (x % P, 1_i64, n);
    while n > 0 {
        if n % 2 != 0 {
            y = (y * x) % P;
            n -= 1;
        }

        x = (x * x) % P;
        n /= 2;
    }
    y
}

pub fn inv(x: i64) -> i64 {
    pow(x, P - 2)
}

fn main() {
    let (N, A, B, C) = read!(usize, i64, i64, i64);

    let M = 2 * N;

    // fact[n] = n! (mod P)
    let mut fact = vec![0; M + 1];
    fact[0] = 1;
    fact[1] = 1;
    for n in 2..M + 1 {
        // n! = (n - 1!) * n
        fact[n] = fact[n - 1] * n as i64 % P;
    }

    // C(n, r) = n! / ((n - r)! r!)
    let combo = |n: usize, r: usize| {
        let mut z = inv(fact[n - r]);
        z *= inv(fact[r]);
        z %= P;
        z *= fact[n];
        z %= P;
        z
    };

    let mut E = 0;
    let percent = inv(100);
    let A = A * percent % P;
    let B = B * percent % P;
    let C = C * percent % P;
    let D = inv(P + 1 - C);
    debug!((A, B, C, D));

    let A = A * D % P;
    let B = B * D % P;

    for x in N..2 * N {
        let mut p = 1;
        p *= pow(A, N as i64);
        p %= P;
        p *= pow((P + 1 - A) % P, (x - N) as i64);
        p %= P;

        let mut q = 1;
        q *= pow(B, N as i64);
        q %= P;
        q *= pow((P + 1 - B) % P, (x - N) as i64);
        p %= P;

        let mut r = p + q;
        r %= P;
        r *= combo(x - 1, N - 1);
        r %= P;
        r *= x as i64;
        r %= P;

        E += r;
        E %= P;
    }

    E *= D;
    E %= P;

    println!("{}", E)
}