


def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   global f  # a global dp table for knapsack
   if f[i][j] < 0:
      if j < wt[i - 1]:
         __a					=  mf_knapsack(i - 1		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )
      else:
         __a					=  max(
             mf_knapsack(i - 1		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )		,				mf_knapsack(i - 1		,				lowerCAmelCase__		,				lowerCAmelCase__		,				j - wt[i - 1]   ) + val[i - 1]		,				)
      __a					=  val
   return f[i][j]




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   __a					=  [[0] * (w + 1) for _ in range(n + 1   )]

   for i in range(1		,				n + 1   ):
      for w_ in range(1		,				w + 1   ):
         if wt[i - 1] <= w_:
            __a					=  max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]		,				dp[i - 1][w_]   )
         else:
            __a					=  dp[i - 1][w_]

   return dp[n][w_], dp




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   if not (isinstance(lowerCAmelCase__		,				(list, tuple)   ) and isinstance(lowerCAmelCase__		,				(list, tuple)   )):
      raise ValueError(
          """Both the weights and values vectors must be either lists or tuples"""   )

   __a					=  len(lowerCAmelCase__   )
   if num_items != len(lowerCAmelCase__   ):
      __a					=  (
          """The number of weights must be the same as the number of values.\n"""
          f'''But got {num_items} weights and {len(lowerCAmelCase__   )} values'''
      )
      raise ValueError(lowerCAmelCase__   )
   for i in range(lowerCAmelCase__   ):
      if not isinstance(wt[i]		,				lowerCAmelCase__   ):
         __a					=  (
             """All weights must be integers but got weight of """
             f'''type {type(wt[i]   )} at index {i}'''
         )
         raise TypeError(lowerCAmelCase__   )

   __a	,    __a					=  knapsack(lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )
   __a					=  set()
   _construct_solution(lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   )

   return optimal_val, example_optional_set




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   # for the current item i at a maximum weight j to be part of an optimal subset,
   # the optimal value at (i, j) must be greater than the optimal value at (i-1, j).
   # where i - 1 means considering only the previous items at the given maximum weight
   if i > 0 and j > 0:
      if dp[i - 1][j] == dp[i][j]:
         _construct_solution(lowerCAmelCase__		,				lowerCAmelCase__		,				i - 1		,				lowerCAmelCase__		,				lowerCAmelCase__   )
      else:
         optimal_set.add(lowerCAmelCase__   )
         _construct_solution(lowerCAmelCase__		,				lowerCAmelCase__		,				i - 1		,				j - wt[i - 1]		,				lowerCAmelCase__   )


if __name__ == "__main__":
  SCREAMING_SNAKE_CASE							    =							[3, 2, 4, 4]
  SCREAMING_SNAKE_CASE							    =							[4, 3, 2, 3]
  SCREAMING_SNAKE_CASE							    =							4
  SCREAMING_SNAKE_CASE							    =							6
  SCREAMING_SNAKE_CASE							    =							[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
  SCREAMING_SNAKE_CASE    , SCREAMING_SNAKE_CASE							    =							knapsack(w, wt, val, n)
  print(optimal_solution)
  print(mf_knapsack(n, wt, val, w))  # switched the n and w

  # testing the dynamic programming problem with example
  # the optimal subset for the above example are items 3 and 4
  SCREAMING_SNAKE_CASE    , SCREAMING_SNAKE_CASE							    =							knapsack_with_example_solution(w, wt, val)
  assert optimal_solution == 8
  assert optimal_subset == {3, 4}
  print('optimal_value = ', optimal_solution)
  print('An optimal subset corresponding to the optimal value', optimal_subset)










'''simple docstring'''




from math import factorial



def 		__A		(  lowerCamelCase_			,			lowerCamelCase_	):


		"""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.""",
							)



