



def        lowercase_							(A : list[int]		,					A : list[int]		,					A : int      ):
       return not any(
           neighbour == 1 and colored_vertices[i] == color
           for i, neighbour in enumerate(A      )      )
def        lowercase_							(A : list[list[int]]		,					A : int		,					A : list[int]		,					A : int      ):

       # Base Case
       if index == len(A      ):
              return True

       # Recursive Step
       for i in range(A      ):
              if valid_coloring(graph[index]		,					A		,					A      ):
                     # Color current vertex
                     snake_case__							:	Tuple   	=	i
                     # Validate coloring
                     if util_color(A		,					A		,					A		,					index + 1      ):
                            return True
                     # Backtrack
                     snake_case__							:	int   	=	-1
       return False





def        lowercase_							(A : list[list[int]]		,					A : int      ):
       snake_case__							:	Union[str, Any]   	=	[-1] * len(A      )

       if util_color(A		,					A		,					A		,					0      ):
              return colored_vertices

       return []











from math import factorial
def        lowercase_							(A : int		,					A : int		,					A : float      ):
       if successes > trials:
              raise ValueError('successes must be lower or equal to trials'      )
       if trials < 0 or successes < 0:
              raise ValueError('the function is defined for non-negative integers'      )
       if not isinstance(A		,					A      ) or not isinstance(A		,					A      ):
              raise ValueError('the function is defined for non-negative integers'      )
       if not 0 < prob < 1:
              raise ValueError('prob has to be in range of 1 - 0'      )
       snake_case__							:	List[Any]   	=	(prob**successes) * ((1 - prob) ** (trials - successes))
       # Calculate the binomial coefficient: n! / k!(n-k)!
       snake_case__							:	List[str]   	=	float(factorial(A      )      )
       coefficient /= factorial(A      ) * factorial(trials - successes      )
       return probability * coefficient


if __name__ == "__main__":
 from doctest import testmod

 testmod()
 print("Probability of 2 successes out of 4 trails")
 print("with probability of 0.75 is:", end=" ")
 print(binomial_distribution(2, 4, 0.75))






