


'''simple docstring'''






def lowerCamelCase__      (						__lowerCamelCase	:							Union[str, Any]							):







   '''simple docstring'''




   _UpperCAmelCase       :       List[str]							   =0
   _UpperCAmelCase       :       List[Any]							   =len(__lowerCamelCase							)

   for i in range(n - 1							):
      for j in range(i + 1	,   __lowerCamelCase							):
         if arr[i] > arr[j]:
            num_inversions += 1

   return num_inversions





def lowerCamelCase__      (						__lowerCamelCase	:							str							):







   '''simple docstring'''




   if len(__lowerCamelCase							) <= 1:
      return arr, 0
   _UpperCAmelCase       :       Optional[Any]							   =len(__lowerCamelCase							) // 2
   _UpperCAmelCase       :       int							   =arr[0:mid]
   _UpperCAmelCase       :       Optional[int]							   =arr[mid:]

   _UpperCAmelCase   ,	_UpperCAmelCase       :       int							   =count_inversions_recursive(__lowerCamelCase							)
   _UpperCAmelCase   ,	_UpperCAmelCase       :       List[Any]							   =count_inversions_recursive(__lowerCamelCase							)
   _UpperCAmelCase   ,	_UpperCAmelCase       :       Optional[Any]							   =_count_cross_inversions(__lowerCamelCase	,   __lowerCamelCase							)

   _UpperCAmelCase       :       Dict							   =inversion_p + inversions_q + cross_inversions
   return c, num_inversions





def lowerCamelCase__      (						__lowerCamelCase	:							List[Any]	,   __lowerCamelCase	:							Optional[Any]							):







   '''simple docstring'''




   _UpperCAmelCase       :       str							   =[]
   _UpperCAmelCase       :       Optional[int]							   =0
   while i < len(__lowerCamelCase							) and j < len(__lowerCamelCase							):
      if p[i] > q[j]:
         # if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
         # These are all inversions. The claim emerges from the
         # property that P is sorted.
         num_inversion += len(__lowerCamelCase							) - i
         r.append(q[j]							)
         j += 1
      else:
         r.append(p[i]							)
         i += 1

   if i < len(__lowerCamelCase							):
      r.extend(p[i:]							)
   else:
      r.extend(q[j:]							)

   return r, num_inversion





def lowerCamelCase__      (						):







   '''simple docstring'''




   _UpperCAmelCase       :       int							   =[1_0, 2, 1, 5, 5, 2, 1_1]

   # this arr has 8 inversions:
   # (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

   _UpperCAmelCase       :       Any							   =count_inversions_bf(__lowerCamelCase							)
   _UpperCAmelCase   ,	_UpperCAmelCase       :       Optional[Any]							   =count_inversions_recursive(__lowerCamelCase							)

   assert num_inversions_bf == num_inversions_recursive == 8

   print('number of inversions = '	,   __lowerCamelCase							)

   # testing an array with zero inversion (a sorted arr_1)

   arr_a.sort()
   _UpperCAmelCase       :       Optional[int]							   =count_inversions_bf(__lowerCamelCase							)
   _UpperCAmelCase   ,	_UpperCAmelCase       :       str							   =count_inversions_recursive(__lowerCamelCase							)

   assert num_inversions_bf == num_inversions_recursive == 0
   print('number of inversions = '	,   __lowerCamelCase							)

   # an empty list should also have zero inversions
   _UpperCAmelCase       :       str							   =[]
   _UpperCAmelCase       :       List[Any]							   =count_inversions_bf(__lowerCamelCase							)
   _UpperCAmelCase   ,	_UpperCAmelCase       :       Tuple							   =count_inversions_recursive(__lowerCamelCase							)

   assert num_inversions_bf == num_inversions_recursive == 0
   print('number of inversions = '	,   __lowerCamelCase							)


if __name__ == "__main__":
 main()





'''simple docstring'''






def lowerCamelCase__      (						__lowerCamelCase	:							int	,   __lowerCamelCase	:							list[int]	,   __lowerCamelCase	:							int							):







   '''simple docstring'''




   def count_of_possible_combinations(__lowerCamelCase	:							int							) -> int:
      if target < 0:
         return 0
      if target == 0:
         return 1
      return sum(count_of_possible_combinations(target - item							) for item in array							)

   return count_of_possible_combinations(__lowerCamelCase							)





def lowerCamelCase__      (						__lowerCamelCase	:							int	,   __lowerCamelCase	:							list[int]	,   __lowerCamelCase	:							int							):







   '''simple docstring'''




   def count_of_possible_combinations_with_dp_array(
       __lowerCamelCase	:							int	,   __lowerCamelCase	:							list[int]							) -> int:
      if target < 0:
         return 0
      if target == 0:
         return 1
      if dp_array[target] != -1:
         return dp_array[target]
      _UpperCAmelCase       :       str							   =sum(
          count_of_possible_combinations_with_dp_array(target - item	,   __lowerCamelCase							)
          for item in array							)
      _UpperCAmelCase       :       Optional[Any]							   =answer
      return answer

   _UpperCAmelCase       :       int							   =[-1] * (target + 1)
   return count_of_possible_combinations_with_dp_array(__lowerCamelCase	,   __lowerCamelCase							)





def lowerCamelCase__      (						__lowerCamelCase	:							int	,   __lowerCamelCase	:							list[int]	,   __lowerCamelCase	:							int							):







   '''simple docstring'''




   _UpperCAmelCase       :       Tuple							   =[0] * (target + 1)
   _UpperCAmelCase       :       Dict							   =1

   for i in range(1	,   target + 1							):
      for j in range(__lowerCamelCase							):
         if i - array[j] >= 0:
            dp_array[i] += dp_array[i - array[j]]

   return dp_array[target]


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 lowercase							=3
 lowercase							=5
 lowercase							=[1, 2, 5]
 print(combination_sum_iv(n, array, target))

