
from math import factorial






def 	a_		(	__lowercase       :     int	, __lowercase       :     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(__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(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.''',
			)

def 	a_		(	__lowercase       :     list[int]	, __lowercase       :     list[int]			)			->  tuple[float, float]:

			# Check if the input is valid
			if not len(__lowercase			) == len(__lowercase			) == 3:
						raise ValueError('Please enter a valid equation.'			)
			if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
						raise ValueError('Both a & b of two equations can\'t be zero.'			)

			# Extract the coefficients
			_snake_case     , _snake_case     , _snake_case          =      equationa
			_snake_case     , _snake_case     , _snake_case          =      equationa

			# Calculate the determinants of the matrices
			_snake_case          =      aa * ba - aa * ba
			_snake_case          =      ca * ba - ca * ba
			_snake_case          =      aa * ca - aa * ca

			# Check if the system of linear equations has a solution (using Cramer's rule)
			if determinant == 0:
						if determinant_x == determinant_y == 0:
									raise ValueError('Infinite solutions. (Consistent system)'			)
						else:
									raise ValueError('No solution. (Inconsistent system)'			)
			else:
						if determinant_x == determinant_y == 0:
									# Trivial solution (Inconsistent system)
									return (0.0, 0.0)
						else:
									_snake_case          =      determinant_x / determinant
									_snake_case          =      determinant_y / determinant
									# Non-Trivial Solution (Consistent system)
									return (x, y)