





from math import factorial







def       UpperCAmelCase							(     _lowerCamelCase      :  int			,   _lowerCamelCase      :  int     ):





						'''simple docstring'''


						if n < k or k < 0:
												raise ValueError("Please enter positive integers for n and k where n >= k"     )
						return factorial(_lowerCamelCase     ) // (factorial(_lowerCamelCase     ) * 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.",
				)






from __future__ import annotations







def       UpperCAmelCase							(     _lowerCamelCase      :  list[int | float]			,   _lowerCamelCase      :  int			,   _lowerCamelCase      :  int     ):





						'''simple docstring'''


						if len(_lowerCamelCase     ) == 0:
												raise ValueError("find_max() arg is an empty sequence"     )
						if (
						    left >= len(_lowerCamelCase     )
						    or left < -len(_lowerCamelCase     )
						    or right >= len(_lowerCamelCase     )
						    or right < -len(_lowerCamelCase     )
						):
												raise IndexError("list index out of range"     )
						if left == right:
												return nums[left]
						SCREAMING_SNAKE_CASE__     :				Optional[int]   =					(left + right) >> 1  # the middle
						SCREAMING_SNAKE_CASE__     :				List[Any]   =					find_max(_lowerCamelCase			,   _lowerCamelCase			,   _lowerCamelCase     )  # find max in range[left, mid]
						SCREAMING_SNAKE_CASE__     :				Optional[int]   =					find_max(_lowerCamelCase			,   mid + 1			,   _lowerCamelCase     )  # find max in range[mid + 1, right]

						return left_max if left_max >= right_max else right_max


if __name__ == "__main__":
				import doctest

				doctest.testmod(verbose=True)