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)


DIV7 = 10**9 + 7
DIV7_5_LIST = (1..31).select do |i|
  (DIV7 - 2) & (1 << (i - 1)) != 0
end

# n! mod (10^9+7)
FACT_MOD7 = []
# {n!}^(-1) mod (10^9+7)
INV_FACT_MOD7 = []

def make_fact_mod7(n)
  # (a * b) mod m = (a mod m) * (b mod m)
  FACT_MOD7[0] = 1
  FACT_MOD7[1] = 1
  (2..n).each do |i|
    FACT_MOD7[i] = (FACT_MOD7[i - 1] * i) % DIV7
  end
end

def make_inv_fact_mod7(n)
  INV_FACT_MOD7[0] = 1
  INV_FACT_MOD7[1] = 1
  (2..n).each do |i|
    INV_FACT_MOD7[i] = (INV_FACT_MOD7[i - 1] * inv_mod(i)) % DIV7
  end
end

def inv_mod(n)
  bq = [1, n]
  (2..31).each do |i|
    bq[i] = (bq[i - 1]**2) % DIV7
  end
  return DIV7_5_LIST.inject(1) do |acc, num|
    (acc * bq[num]) % DIV7
  end
end

def permutation(height, width)
  return FACT_MOD7[height + width - 2] *
         INV_FACT_MOD7[height - 1] *
         INV_FACT_MOD7[width - 1] %
         DIV7
end

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

make_fact_mod7(h + w)
make_inv_fact_mod7([h, w].max)

total = ((b + 1)..w).map do |x|
  permutation(h - a, x) * permutation(a, (w - x + 1))
end.inject(&:+)
puts total % DIV7