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)

# this is copy from https://atcoder.jp/contests/abc042/submissions/2435693
def create_fact_mod(num_max, mod)
  $fmod = Array.new(num_max+1)
  $fmod[0]=1
  $fmod[1]=1
  (2..num_max).each do |i|
    $fmod[i] = ($fmod[i-1] * i)%mod
  end
end

def create_seq_inv(num_max, mod)
  $simod = Array.new(num_max+1)
  $simod[0]=1
  $simod[1]=1
  (2..num_max).each do |i|
    $simod[i] = (mod-mod/i) * $simod[mod%i] % mod
  end
end

def create_fact_inv_mod(num_max, mod)
  $fimod = Array.new(num_max+1)
  $fimod[0] = 1
  $fimod[1] = 1
  (2..num_max).each do |i|
    $fimod[i] = ($fimod[i-1] * $simod[i]) % mod
  end
end

def comb_mod(n, k, mod)
  $fmod[n] * $fimod[n-k] * $fimod[k] %mod
end

h, w, a, b = gets.strip.split.map(&:to_i)
MOD = 10**9+7


create_fact_mod(h+w, MOD)
create_seq_inv(h+w, MOD)
create_fact_inv_mod(h+w, MOD)

sum = 0
i=1
loop do
  ww = b+i
  hh = a+i
  break if ww > w || hh > h

  c1 = comb_mod(h-hh+ww-1, ww-1, MOD)
  c2 = comb_mod(hh+w-ww-1, hh-1, MOD)
  sum += c1*c2 % MOD
  sum %= MOD
  i += 1
end

p sum