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 (1.15.1) (judge verdict: Accepted)

#![allow(non_snake_case)]
#![allow(unused_variables)]
#![allow(dead_code)]

fn main() {
    let (H, W, A, B): (usize, usize, usize, usize) = {
        let mut line: String = String::new();
        std::io::stdin().read_line(&mut line).unwrap();
        let mut iter = line.split_whitespace();
        (
            iter.next().unwrap().parse().unwrap(),
            iter.next().unwrap().parse().unwrap(),
            iter.next().unwrap().parse().unwrap(),
            iter.next().unwrap().parse().unwrap(),
        )
    };

    const MOD: i64 = 1_000_000_007;
    fn modinv(x: i64) -> i64 {
        let mut a = x;
        let mut b = MOD;
        let mut u = 1;
        let mut v = 0;
        while b != 0 {
            let t = a / b;
            a -= t * b;
            std::mem::swap(&mut a, &mut b);
            u -= t * v;
            std::mem::swap(&mut u, &mut v);
        }
        u %= MOD;
        if u >= 0 {
            u
        } else {
            u + MOD
        }
    }
    #[derive(Copy, Clone)]
    struct ModInt {
        x: i64,
    }
    impl ModInt {
        fn new(x: i64) -> ModInt {
            let y = if x >= 0 {
                x % MOD
            } else {
                (x + (1 - x / MOD) * MOD) % MOD
            };
            ModInt { x: y }
        }
    }
    impl std::ops::Add<ModInt> for ModInt {
        type Output = ModInt;
        fn add(self, other: ModInt) -> ModInt {
            ModInt::new(self.x + other.x)
        }
    }
    impl std::ops::Add<i64> for ModInt {
        type Output = ModInt;
        fn add(self, other: i64) -> ModInt {
            ModInt::new(self.x + other)
        }
    }
    impl std::ops::Add<usize> for ModInt {
        type Output = ModInt;
        fn add(self, other: usize) -> ModInt {
            ModInt::new(self.x + (other as i64))
        }
    }
    impl std::ops::Sub<ModInt> for ModInt {
        type Output = ModInt;
        fn sub(self, other: ModInt) -> ModInt {
            ModInt::new(self.x - other.x)
        }
    }
    impl std::ops::Sub<i64> for ModInt {
        type Output = ModInt;
        fn sub(self, other: i64) -> ModInt {
            ModInt::new(self.x - other)
        }
    }
    impl std::ops::Sub<usize> for ModInt {
        type Output = ModInt;
        fn sub(self, other: usize) -> ModInt {
            ModInt::new(self.x - (other as i64))
        }
    }
    impl std::ops::Mul<ModInt> for ModInt {
        type Output = ModInt;
        fn mul(self, other: ModInt) -> ModInt {
            ModInt::new(self.x * other.x)
        }
    }
    impl std::ops::Mul<i64> for ModInt {
        type Output = ModInt;
        fn mul(self, other: i64) -> ModInt {
            ModInt::new(self.x * other)
        }
    }
    impl std::ops::Mul<usize> for ModInt {
        type Output = ModInt;
        fn mul(self, other: usize) -> ModInt {
            ModInt::new(self.x * (other as i64))
        }
    }
    impl std::ops::Div<ModInt> for ModInt {
        type Output = ModInt;
        fn div(self, other: ModInt) -> ModInt {
            ModInt::new(self.x * modinv(other.x as i64))
        }
    }
    impl std::ops::Div<i64> for ModInt {
        type Output = ModInt;
        fn div(self, other: i64) -> ModInt {
            ModInt::new(self.x * modinv(other))
        }
    }
    impl std::ops::Div<usize> for ModInt {
        type Output = ModInt;
        fn div(self, other: usize) -> ModInt {
            ModInt::new(self.x * modinv(other as i64))
        }
    }
    impl std::fmt::Display for ModInt {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(f, "{}", self.x)
        }
    }

    let fact: Vec<_> = (0..((H + W) + 1))
        .scan(ModInt::new(1), |f, i| {
            *f = if i == 0 { ModInt::new(1) } else { *f * i };
            Some(*f)
        })
        .collect();
    fn comb(n: usize, r: usize, fact: &Vec<ModInt>) -> ModInt {
        fact[n] / (fact[r] * fact[n - r])
    }

    let ans = (0..(W - B))
        .map(|i| {
            comb(H - A - 1 + B + i, H - A - 1, &fact) * comb(A - 1 + W - B - 1 - i, A - 1, &fact)
        })
        .fold(ModInt::new(0), |res, x| res + x);

    println!("{}", ans);
}