#! /usr/bin/python3
# Last edited on 2026-09-08 17:21:33 by stolfi

import math, sys
from math import comb

def prob(N, C, M, F):

  # Consider a set of N boxes, C of which contain one coin each. 
  # This function computes the probability that picking M
  # of the N boxes at random will yield F of the C coins.
  # That is comb(C,F)*comb(N-C,M-F)/comb(N,M).
  
  # Boundary logic checks
  if F < 0 or F > C or F > M or (M - F) > (N - C):
    return 0.0

  def ln_fact(n):
    # Computes ln(n!) safely.
    return math.lgamma(n + 1)
    
  def ln_comb(n,k):
    # Computes ln(comb(n,k)) safely.
    assert n > 0 and k >= 0 and k <= n
    return ln_fact(n) - ln_fact(k) - ln_fact(n-k)

  # Combine the log pieces and convert back via exponentiation
  ln_prob = ln_comb(C,F) + ln_comb(N-C,M-F) - ln_comb(N,M)
  return math.exp(ln_prob)

assert abs(prob(10,1,1,1) - 1/10) < 1.0e-6
assert abs(prob(10,3,1,1) - 3/10) < 1.0e-6
assert abs(prob(10,3,2,1) - 42/90) < 1.0e-6
assert abs(prob(10,3,2,2) - 6/90) < 1.0e-6
ia = 1
N = int(sys.argv[ia]); ia += 1
C = int(sys.argv[ia]); ia += 1
M = int(sys.argv[ia]); ia += 1
F = int(sys.argv[ia]); ia += 1
assert ia == len(sys.argv)

# if ln_comb(M,F) < ln_comb(MAX_INT):
print("comb(%d,%d) = %d" % (M, F, comb(M,F)))
P = prob(N,C,M,F)
print("P = %.30f" % P)
print("1/P = %.0f" % (1/P))