





import math
import sys




def 		lowerCAmelCase_    (  __UpperCAmelCase:			int      )      ->	int:
    if number != int(__UpperCAmelCase      ):
        raise ValueError('''the value of input must be a natural number'''      )
    if number < 0:
        raise ValueError('''the value of input must not be a negative number'''      )
    if number == 0:
        return 1
    UpperCamelCase__							:				Any          =      [-1] * (number + 1)
    UpperCamelCase__							:				Any          =      0
    for i in range(1		,   number + 1      ):
        UpperCamelCase__							:				Tuple          =      sys.maxsize
        UpperCamelCase__							:				Optional[int]          =      int(math.sqrt(__UpperCAmelCase      )      )
        for j in range(1		,   root + 1      ):
            UpperCamelCase__							:				Tuple          =      1 + answers[i - (j**2)]
            UpperCamelCase__							:				List[Any]          =      min(__UpperCAmelCase		,   __UpperCAmelCase      )
        UpperCamelCase__							:				str          =      answer
    return answers[number]


if __name__ == "__main__":
  import doctest

  doctest.testmod()








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.',
  )

