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 (2.3.3) (judge verdict: Accepted)

require 'openssl'; BN = OpenSSL::BN
M = BN.new(10**9+7)
H, W, A, B = gets.split.map(&:to_i)

class NCR
  @fac = []
  class << self
    def init(n); calc_fact(n); end
    def calc_fact(n)
      @fac[0] = BN.new(1)
      @fac[1] = BN.new(1)
      i = 2
      while i <= n
        @fac[i] = @fac[i-1].mod_mul(i, M)
        i += 1
      end
    end
    def nCr(n, r)
       @fac[n].mod_mul( @fac[r].mod_inverse(M).mod_mul( @fac[n-r].mod_inverse(M), M), M)
    end
  end
end

NCR.init(H+W)

def n_pats(h, w)
  NCR.nCr(h+w, h)
end

mid_row = Array.new(W){|x|
  n_pats(H-A-1, x) if x >= B
}

last_row = Array.new(W){|x|
  mid_row[x].mod_mul( n_pats(A-1, W-x-1), M) if x >= B
}

p last_row.compact.inject{|a, b| a.mod_add(b, M) }.to_i