





from math import factorial



def A_      (						A__						, A__							)    ->						int:

       if n < k or k < 0:
              raise ValueError('Please enter positive integers for n and k where n >= k'							)
       return factorial(lowercase_							) // (factorial(lowercase_							) * factorial(n - k							))


if __name__ == "__main__":
    print(
        """The number of five-card hands possible from a standard""",
        F"""fifty-two card deck is: {combinations(5_2, 5)}\n""",
    )

    print(
        """If a class of 40 students must be arranged into groups of""",
        F"""4 for group projects, there are {combinations(4_0, 4)} ways""",
        """to arrange them.\n""",
    )

    print(
        """If 10 teams are competing in a Formula One race, there""",
        F"""are {combinations(1_0, 3)} ways that first, second and""",
        """third place can be awarded.""",
    )









"""simple docstring"""


from collections.abc import Callable

import numpy as np




def        lowercase__     (       lowercase_		,lowercase_		,lowercase_		,lowercase_		,lowercase_      )       ->		np.ndarray:




      """simple docstring"""

      _UpperCamelCase   :						List[str]    =							int(np.ceil((x_end - xa) / step_size      )      )
      _UpperCamelCase   :						Tuple    =							np.zeros((n + 1,)      )
      _UpperCamelCase   :						List[str]    =							ya
      _UpperCamelCase   :						List[Any]    =							xa

      for k in range(lowercase_      ):
            _UpperCamelCase   :						List[str]    =							y[k] + step_size * ode_func(lowercase_		,y[k]      )
            x += step_size

      return y


if __name__ == "__main__":
  import doctest

  doctest.testmod()
