





from itertools import product




def 						A_			(  lowercase_ = 15  )			->						int:
       _snake_case      :							float     				=						0.0
       _snake_case      :							float
       _snake_case      :							int
       _snake_case      :							int
       _snake_case      :							int
       _snake_case      :							int
       _snake_case      :							tuple[int, ...]

       for series in product(range(2  )      ,		repeat=lowercase_  ):
              _snake_case      :							Any     				=						series.count(1  )
              _snake_case      :							List[Any]     				=						num_turns - num_blue
              if num_red >= num_blue:
                     continue
              _snake_case      :							Union[str, Any]     				=						1.0
              for ind, col in enumerate(lowercase_      ,		2  ):
                     if col == 0:
                            prob *= (ind - 1) / ind
                     else:
                            prob *= 1 / ind

              total_prob += prob

       return int(1 / total_prob  )


if __name__ == "__main__":
   print(F"""{solution() = }""")













import numpy as np




def 						A_			(  lowercase_  )			->						Optional[Any]:
       _snake_case      :							Union[str, Any]     				=						np.exp(lowercase_  )

       # Add up the all the exponentials
       _snake_case      :							List[str]     				=						np.sum(lowercase_  )

       # Divide every exponent by the sum of all exponents
       _snake_case      :							Optional[int]     				=						exponent_vector / sum_of_exponents

       return softmax_vector


if __name__ == "__main__":
   print(softmax((0,)))






