



'''simple docstring'''

import math



def      __lowerCamelCase     (      lowerCAmelCase_  )			->      bool:
		assert isinstance(__UpperCAmelCase						,			__UpperCAmelCase  ) and (
		    number >= 0
		), "'number' must been an int and positive"

		if 1 < number < 4:
				# 2 and 3 are primes
				return True
		elif number < 2 or not number % 2:
				# Negatives, 0, 1 and all even numbers are not primes
				return False

		_a     :      Optional[Any]        =							range(3						,			int(math.sqrt(__UpperCAmelCase  ) + 1  )						,			2  )
		return not any(not number % i for i in odd_numbers  )



def      __lowerCamelCase     (      lowerCAmelCase_						,			lowerCAmelCase_=1						,			**lowerCAmelCase_  )			->      Union[str, Any]:
		_a     :      Tuple        =							factor * value
		_a     :      List[str]        =							value

		while not is_prime(__UpperCAmelCase  ):
				value += 1 if not ("desc" in kwargs and kwargs["desc"] is True) else -1

		if value == first_value_val:
				return next_prime(value + 1						,			**__UpperCAmelCase  )
		return value











from math import factorial




def 		lowerCAmelCase_    (  __UpperCAmelCase:			int		,   __UpperCAmelCase:			int      )      ->	int:

    # If either of the conditions are true, the function is being asked
    # to calculate a factorial of a negative number, which is not possible
    if n < k or k < 0:
        raise ValueError('''Please enter positive integers for n and k where n >= k'''      )
    return factorial(__UpperCAmelCase      ) // (factorial(__UpperCAmelCase      ) * factorial(n - k      ))


if __name__ == "__main__":
  print(
      'The number of five-card hands possible from a standard',
      F'''fifty-two card deck is: {combinations(52, 5)}\n''',
  )

  print(
      'If a class of 40 students must be arranged into groups of',
      F'''4 for group projects, there are {combinations(40, 4)} ways''',
      'to arrange them.\n',
  )

  print(
      'If 10 teams are competing in a Formula One race, there',
      F'''are {combinations(10, 3)} ways that first, second and''',
      'third place can be awarded.',
  )

