





"""simple docstring"""






from math import factorial






def      lowercase__ (					lowerCAmelCase__   :			Any							,			lowerCAmelCase__   :			Tuple			)    ->			Optional[int]:



			'''simple docstring'''




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


   """simple docstring"""




   lowercase_						:     Optional[int]							     =  2
   lowercase_						:     Tuple							     =  []
   while i * i <= n:
      if n % i:
         i += 1
      else:
         n //= i
         factors.append(_UpperCamelCase				)
   if n > 1:
      factors.append(_UpperCamelCase				)
   return factors


if __name__ == "__main__":
 import doctest

 doctest.testmod()




