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 Ruby (2.7.1) (judge verdict: Time Limit Exceeded)

PRIME = 10 ** 9 + 7
class Integer
  @@memo = Hash.new{nil}
  def fac
    return 1 if self.zero?
    ary = (1..self).to_a
    loop do
      ary = ary.each_slice(2).map do |a, b|
        b ? a * b % PRIME : a
      end
      return ary[0] if ary.size == 1
    end
  end

  def combi r
    self.fac * (r.fac * (self - r).fac).modinv(PRIME) % PRIME
  end

  def modinv mod
    return @@memo[self] if @@memo[self]
    pow = mod - 2
    factors = []
    until pow.zero? do
      tmp = Math.log(pow, 2).floor
      factors.push tmp
      pow -= 2 ** tmp
    end
    factors.reverse!

    prev = self ** (2 ** factors[0])
    ans = factors.each_with_index.inject(1) do |res, (i, idx)|
      res = if idx.zero?
        res * prev % mod
      else
        prev = prev ** (2 ** (i - factors[idx - 1])) % mod
        res * prev % mod
      end
      res
    end
    @@memo[self] = ans
  end
end


h, w, a, b = gets.chomp.split.map(&:to_i)

s = (w - b - 1 + a - 1).combi(w - b - 1); 0
point = (b + h - a - 1).combi(b); 0
ans = (b..(w - 1)).each_with_index.reduce(0) do |sum, (i, idx)|
  #p "sum: #{sum}, idx:#{idx}, i:#{i}"
  sum = (sum + point * s) % PRIME
  s = s * (w - b - 1 - idx) * (w - b - 1 + a - 1 - idx).modinv(PRIME) % PRIME rescue nil
  point = point * (i + 1 + h - a - 1) * (i + 1).modinv(PRIME) % PRIME
  sum
end
puts ans % PRIME