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)

#[allow(unused_imports)]
use {
   proconio::{fastout, input, marker::*},
   std::cmp::*,
   std::collections::*,
};

#[fastout]
fn main() {
   input! {
      h: Usize1,
      w: Usize1,
      a: usize,
      b: usize,
   }
   let c = Comb::new(h + w);
   let mut ans = Mint::new(0);
   let mut x = h - a;
   let mut y = b;
   loop {
      ans += c.c(x + y, x) * c.c(h + w - x - y, h - x);
      if x == 0 || y == w {
         break;
      }
      x -= 1;
      y += 1;
   }
   println!("{}", ans);
}

const MOD: i64 = 1000000007;
#[derive(Copy, Clone)]
pub struct Mint {
   x: i64,
}
impl Mint {
   fn new(x: i64) -> Self {
      Mint {
         x: (x % MOD + MOD) % MOD,
      }
   }
   fn pow(self, n: usize) -> Self {
      match n {
         0 => Mint::new(1),
         _ => {
            let mut a = self.pow(n >> 1);
            a *= a;
            if n & 1 == 1 {
               a *= self;
            }
            a
         }
      }
   }
   fn inv(self) -> Self { self.pow((MOD - 2) as usize) }
}
impl std::ops::Neg for Mint {
   type Output = Mint;
   fn neg(self) -> Self::Output { Self::new(-self.x) }
}
impl std::ops::AddAssign<Mint> for Mint {
   fn add_assign(&mut self, rhs: Self) {
      self.x += rhs.x;
      if self.x >= MOD {
         self.x -= MOD;
      }
   }
}
impl std::ops::AddAssign<i64> for Mint {
   fn add_assign(&mut self, rhs: i64) { *self += Mint::new(rhs); }
}
impl<T> std::ops::Add<T> for Mint
where
   Mint: std::ops::AddAssign<T>,
{
   type Output = Self;
   fn add(self, other: T) -> Self {
      let mut res = self;
      res += other;
      res
   }
}
impl std::ops::SubAssign<Mint> for Mint {
   fn sub_assign(&mut self, rhs: Self) {
      self.x += MOD - rhs.x;
      if self.x >= MOD {
         self.x -= MOD;
      }
   }
}
impl std::ops::SubAssign<i64> for Mint {
   fn sub_assign(&mut self, rhs: i64) { *self -= Mint::new(rhs); }
}
impl<T> std::ops::Sub<T> for Mint
where
   Mint: std::ops::SubAssign<T>,
{
   type Output = Self;
   fn sub(self, other: T) -> Self {
      let mut res = self;
      res -= other;
      res
   }
}
impl std::ops::MulAssign<Mint> for Mint {
   fn mul_assign(&mut self, rhs: Self) {
      self.x *= rhs.x;
      self.x %= MOD;
   }
}
impl std::ops::MulAssign<i64> for Mint {
   fn mul_assign(&mut self, rhs: i64) { *self *= Mint::new(rhs); }
}
impl<T> std::ops::Mul<T> for Mint
where
   Mint: std::ops::MulAssign<T>,
{
   type Output = Self;
   fn mul(self, other: T) -> Self {
      let mut res = self;
      res *= other;
      res
   }
}
impl std::ops::DivAssign<Mint> for Mint {
   fn div_assign(&mut self, rhs: Self) { *self *= rhs.inv(); }
}
impl std::ops::DivAssign<i64> for Mint {
   fn div_assign(&mut self, rhs: i64) { *self *= Mint::new(rhs).inv(); }
}
impl<T> std::ops::Div<T> for Mint
where
   Mint: std::ops::DivAssign<T>,
{
   type Output = Self;
   fn div(self, other: T) -> Self {
      let mut res = self;
      res /= other;
      res
   }
}
impl<T> From<T> for Mint
where
   i64: From<T>,
{
   fn from(x: T) -> Mint { Mint::new(i64::from(x)) }
}
impl std::fmt::Display for Mint {
   fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { write!(f, "{}", self.x) }
}
struct Comb {
   n: usize,
   fact: Vec<Mint>,
   ifact: Vec<Mint>,
}
impl Comb {
   fn new(n: usize) -> Self {
      let mut fact = vec![Mint::new(1); n + 1];
      let mut ifact = vec![Mint::new(1); n + 1];
      for i in 1..n + 1 {
         fact[i] = fact[i - 1] * i as i64;
      }
      ifact[n] = fact[n as usize].inv();
      for i in (1..n + 1).rev() {
         ifact[i - 1] = ifact[i] * i as i64;
      }
      Comb {
         n: n,
         fact: fact,
         ifact: ifact,
      }
   }
   fn c(&self, n: usize, r: usize) -> Mint {
      if r > n || n > self.n {
         Mint::new(0)
      } else {
         self.fact[n] * self.ifact[r] * self.ifact[n - r]
      }
   }
}