Problem:
Score : 500 points

Problem Statement

Takahashi and Aoki will play a game. They will repeatedly play it until one of them have N wins in total.

When they play the game once, Takahashi wins with probability A %, Aoki wins with probability B %, and the game ends in a draw (that is, nobody wins) with probability C %.
Find the expected number of games that will be played, and print it as follows.

We can represent the expected value as P/Q with coprime integers P and Q.
Print the integer R between 0 and 10^9+6 (inclusive) such that R \times Q \equiv P\pmod {10^9+7}.
(Such an integer R always uniquely exists under the constraints of this problem.)

Constraints

1 \leq N \leq 100000

0 \leq A,B,C \leq 100

1 \leq A+B

A+B+C=100

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N A B C

Output

Print the expected number of games that will be played, in the manner specified in the statement.

Sample Input 1

1 25 25 50

Sample Output 1

2

Since N=1, they will repeat the game until one of them wins.
The expected number of games played is 2.

Sample Input 2

4 50 50 0

Sample Output 2

312500008

C may be 0.

Sample Input 3

1 100 0 0

Sample Output 3

1

B may also be 0.

Sample Input 4

100000 31 41 28

Sample Output 4

104136146

Solution in Ruby (2.3.3) (judge verdict: Runtime Error)

N, A, B, C = gets.chop.split.map(&:to_i)

def extgcd(a, b)
  if b.zero?
    {x: 1, y: 0, gcd: a}
  else
    {
      x: prev[:y],
      y: prev[:x] - (a / b) * prev[:y],
      gcd: prev[:gcd]
    }
  end
end

num = Rational(0,1)
(N..(2*N-1)).each do |i|
  comb_a = 1
  comb_b =1
  ((N+1)..i).each do |n|
    comb_a *= n
    comb_b *= n-N
  end
  comb = comb_a / comb_b

  a = A**N * (100-B)**(i-N)
  b = B**N * (100-A)**(i-N)
  num += Rational((a+b)*comb, 100**i)
end

mod_num = (10**9)+7
p extgcd(num.denominator, mod_num)[:x] * num.numerator