



'''simple docstring'''





def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       )      ->      Optional[Any]:
  __lowerCamelCase       :       Optional[int]           = [1]
  for i in range(2       ,_lowerCAmelCase       ):
    factorials.append(factorials[-1] * i       )
  assert 0 <= k < factorials[-1] * n, "k out of bounds"

  __lowerCamelCase       :       Optional[int]           = []
  __lowerCamelCase       :       Optional[Any]           = list(range(_lowerCAmelCase       )       )

  # Find permutation
  while factorials:
    __lowerCamelCase       :       Optional[int]           = factorials.pop()
    __lowerCamelCase       ,__lowerCamelCase       :       Union[str, Any]           = divmod(_lowerCAmelCase       ,_lowerCAmelCase       )
    permutation.append(elements[number]       )
    elements.remove(elements[number]       )
  permutation.append(elements[0]       )

  return permutation


if __name__ == "__main__":
   import doctest

   doctest.testmod()











'''simple docstring'''





from __future__ import annotations

import math
from collections.abc import Callable

def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       ,_lowerCAmelCase       ,_lowerCAmelCase = 100       ,)      ->      float:
  __lowerCamelCase       :       Dict           = x_start
  __lowerCamelCase       :       int           = fnc(_lowerCAmelCase       )
  __lowerCamelCase       :       Dict           = 0.0

  for _ in range(_lowerCAmelCase       ):
    # Approximates curve as a sequence of linear lines and sums their length
    __lowerCamelCase       :       List[str]           = (x_end - x_start) / steps + xa
    __lowerCamelCase       :       List[Any]           = fnc(_lowerCAmelCase       )
    length += math.hypot(xa - xa       ,fxa - fxa       )

    # Increment step
    __lowerCamelCase       :       Any           = xa
    __lowerCamelCase       :       Tuple           = fxa

  return length


if __name__ == "__main__":

   def        a_     (							_lowerCAmelCase       )      ->      Dict:
     return math.sin(10 * x       )

   print('f(x) = sin(10 * x)')
   print('The length of the curve from x = -10 to x = 10 is:')
   _UpperCamelCase											=						10
   while i <= 100000:
      print(f'''With {i} steps: {line_length(f, -10, 10, i)}''')
      i *= 10






