

'''simple docstring'''
def 	_lowerCAmelCase (						_UpperCamelCase						:					int = 60_08_51_47_51_43    )				->      int:
     """simple docstring"""




     try:
          _SCREAMING_SNAKE_CASE							   =int(_UpperCamelCase    )
     except (TypeError, ValueError):
          raise TypeError('Parameter n must be int or castable to int.'    )
     if n <= 0:
          raise ValueError('Parameter n must be greater than or equal to one.'    )
     _SCREAMING_SNAKE_CASE							   =1
     _SCREAMING_SNAKE_CASE							   =2
     while i * i <= n:
          while n % i == 0:
               _SCREAMING_SNAKE_CASE							   =i
               n //= i
          i += 1
     if n > 1:
          _SCREAMING_SNAKE_CASE							   =n
     return int(_UpperCamelCase    )


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



'''simple docstring'''
from math import factorial


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					float    )				->      float:
     """simple docstring"""




     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(_UpperCamelCase							,			_UpperCamelCase    ) or not isinstance(_UpperCamelCase							,			_UpperCamelCase    ):
          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'    )
     _SCREAMING_SNAKE_CASE							   =(prob**successes) * ((1 - prob) ** (trials - successes))
     # Calculate the binomial coefficient: n! / k!(n-k)!
     _SCREAMING_SNAKE_CASE							   =float(factorial(_UpperCamelCase    )    )
     coefficient /= factorial(_UpperCamelCase    ) * 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, 0.7_5))
