Problem:
Score : 400 points

Problem Statement

We have a large square grid with H rows and W columns.
Iroha is now standing in the top-left cell.
She will repeat going right or down to the adjacent cell, until she reaches the bottom-right cell.

However, she cannot enter the cells in the intersection of the bottom A rows and the leftmost B columns. (That is, there are A×B forbidden cells.) There is no restriction on entering the other cells.

Find the number of ways she can travel to the bottom-right cell.

Since this number can be extremely large, print the number modulo 10^9+7.

Constraints

1 ≦ H, W ≦ 100,000

1 ≦ A < H

1 ≦ B < W

Input

The input is given from Standard Input in the following format:

H W A B

Output

Print the number of ways she can travel to the bottom-right cell, modulo 10^9+7.

Sample Input 1

2 3 1 1

Sample Output 1

2

We have a 2×3 grid, but entering the bottom-left cell is forbidden. The number of ways to travel is two: "Right, Right, Down" and "Right, Down, Right".

Sample Input 2

10 7 3 4

Sample Output 2

3570

There are 12 forbidden cells.

Sample Input 3

100000 100000 99999 99999

Sample Output 3

1

Sample Input 4

100000 100000 44444 55555

Sample Output 4

738162020

Solution in Rust (1.42.0) (judge verdict: Accepted)

// -*- coding:utf-8-unix -*-

use num_traits::{pow, One};
use proconio::{fastout, input};
use std::ops::{Add, Div, Mul, Sub};

const MODULUS: usize = 1000000007;

// ABC086C - Traveling
// https://atcoder.jp/contests/abs/fasks/arc089_a

#[fastout]
fn main() {
    input! {
        h :usize,
        w :usize,
        a :usize,
        b :usize,
    }

    // prepare a table of factorials
    let mut fac_vec: Vec<ModP> = Vec::new();
    let mut fac_inv_vec: Vec<ModP> = Vec::new();
    fac_vec.push(ModP(1));
    fac_inv_vec.push(ModP(1));
    for i in 1..=h+w {
        fac_vec.push(ModP(i) * fac_vec[i - 1]);
        fac_inv_vec.push(ModP(1) / fac_vec[i]);
    }
    let fac_vec = fac_vec;
    let fac_inv_vec = fac_inv_vec;

    let mut total = ModP(0);
    for j in b + 1..=w {
        let tate1 = ModP(h - a - 1);
        let yoko1 = ModP(j - 1);
        let tate2 = ModP(a - 1);
        let yoko2 = ModP(w - j);
        total = total
            + combination(tate1 + yoko1, tate1, &fac_vec, &fac_inv_vec)
                * combination(tate2 + yoko2, tate2, &fac_vec, &fac_inv_vec);
    }
    println!("{}", total.0);
}

fn combination(n: ModP, r: ModP, fac_vec: &Vec<ModP>, fac_inv_vec: &Vec<ModP>) -> ModP {
    return fac_vec[n.0] * fac_inv_vec[(n - r).0] * fac_inv_vec[r.0];
}

#[derive(Clone, Copy, PartialEq, Debug)]
struct ModP(usize);

impl One for ModP {
    fn one() -> Self {
        return ModP(1);
    }
}
impl Add for ModP {
    type Output = Self;
    fn add(self, rhs: Self) -> Self {
        return ModP((self.0 + rhs.0) % MODULUS);
    }
}
impl Sub for ModP {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self {
        return ModP((self.0 + MODULUS - rhs.0) % MODULUS);
    }
}
impl Mul for ModP {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self {
        return ModP((self.0 * rhs.0) % MODULUS);
    }
}
impl Div for ModP {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        if rhs.0 == 0 {
            panic!("Tried to divide by ModP(0)!");
        }
        let rhs_inv = pow(rhs, MODULUS - 2);
        return self * rhs_inv;
    }
}