





"""simple docstring"""






from math import factorial






def      lowercase__ (					lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int			)    ->			int:



			'''simple docstring'''

			# 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(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.''',
		)






"""simple docstring"""






from __future__ import annotations






def      lowercase__ (					lowerCAmelCase__   :			int			)    ->			list[int]:



			'''simple docstring'''
			a__  :	Tuple		 =			[True] * limit
			a__  :	Tuple		 =			False
			a__  :	int		 =			False
			a__  :	Tuple		 =			True

			for i in range(3							,			int(limit**0.5 + 1			)							,			2			):
						a__  :	int		 =			i * 2
						while index < limit:
									a__  :	Union[str, Any]		 =			False
									a__  :	int		 =			index + i

			a__  :	str		 =			[2]

			for i in range(3							,			lowerCAmelCase__							,			2			):
						if is_prime[i]:
									primes.append(lowerCAmelCase__			)

			return primes






def      lowercase__ (					lowerCAmelCase__   :			int = 1_0_0_0_0_0_0			)    ->			int:



			'''simple docstring'''
			a__  :	List[Any]		 =			prime_sieve(lowerCAmelCase__			)
			a__  :	int		 =			0
			a__  :	str		 =			0

			for i in range(len(lowerCAmelCase__			)			):
						for j in range(i + length							,			len(lowerCAmelCase__			)			):
									a__  :	List[str]		 =			sum(primes[i:j]			)
									if sol >= ceiling:
												break

									if sol in primes:
												a__  :	Tuple		 =			j - i
												a__  :	Union[str, Any]		 =			sol

			return largest


if __name__ == "__main__":
		print(f"{solution() = }")