





def   a_   (			lowerCAmelCase_				:      Union[str, Any],     lowerCAmelCase_				:      Optional[int],     lowerCAmelCase_				:      Union[str, Any],     lowerCAmelCase_				:      Any,     lowerCAmelCase_				:      Any,     lowerCAmelCase_				:      str     ):
   if index == r:
      for j in range(lowerCAmelCase_     ):
         print(data[j],     end=' '     )
      print(' '     )
      return
   #  When no more elements are there to put in data[]
   if i >= n:
      return
   # current is included, put next at next location
   __lowerCAmelCase							     =     arr[i]
   combination_util(lowerCAmelCase_,     lowerCAmelCase_,     lowerCAmelCase_,     index + 1,     lowerCAmelCase_,     i + 1     )
   # current is excluded, replace it with
   # next (Note that i+1 is passed, but
   # index is not changed)
   combination_util(lowerCAmelCase_,     lowerCAmelCase_,     lowerCAmelCase_,     lowerCAmelCase_,     lowerCAmelCase_,     i + 1     )
   # The main function that prints all combinations
   # of size r in arr[] of size n. This function
   # mainly uses combinationUtil()

def   a_   (			lowerCAmelCase_				:      Any,     lowerCAmelCase_				:      int,     lowerCAmelCase_				:      Optional[Any]     ):
   # A temporary array to store all combination one by one
   __lowerCAmelCase							     =     [0] * r
   # Print all combination using temporary array 'data[]'
   combination_util(lowerCAmelCase_,     lowerCAmelCase_,     lowerCAmelCase_,     0,     lowerCAmelCase_,     0     )


if __name__ == "__main__":
     # Driver code to check the function above
     _snake_case  :   Tuple						   =  [10, 20, 30, 40, 50]
     print_combination(arr, len(arr), 3)
     # This code is contributed by Ambuj sahu








import mpmath  # for roots of unity
import numpy as np



class     _UpperCAmelCase   :

  """simple docstring"""


  def __init__(    self			:		List[Any] ,  lowerCAmelCase_			:		Dict=None ,  lowerCAmelCase_			:		str=None					)							->						List[Any]:
     # Input as list
     __lowerCAmelCase							     =     list(poly_a or [0]					)[:]
     __lowerCAmelCase							     =     list(poly_b or [0]					)[:]

     # Remove leading zero coefficients
     while self.polyA[-1] == 0:
        self.polyA.pop()
     __lowerCAmelCase							     =     len(self.polyA					)

     while self.polyB[-1] == 0:
        self.polyB.pop()
     __lowerCAmelCase							     =     len(self.polyB					)

     # Add 0 to make lengths equal a power of 2
     __lowerCAmelCase							     =     int(
         2 ** np.ceil(np.loga(len(self.polyA					) + len(self.polyB					) - 1					)					)					)

     while len(self.polyA					) < self.c_max_length:
        self.polyA.append(0					)
     while len(self.polyB					) < self.c_max_length:
        self.polyB.append(0					)
     # A complex root used for the fourier transform
     __lowerCAmelCase							     =     complex(mpmath.root(x=1 ,  n=self.c_max_length ,  k=1					)					)

     # The product
     __lowerCAmelCase							     =     self.__multiply()






  def     lowercase		(    self			:		Optional[int] ,  lowerCAmelCase_			:		str					)							->						Optional[int]:
     __lowerCAmelCase							     =     [[x] for x in self.polyA] if which == 'A' else [[x] for x in self.polyB]
     # Corner case
     if len(lowerCAmelCase_					) <= 1:
        return dft[0]
     #
     __lowerCAmelCase							     =     self.c_max_length // 2
     while next_ncol > 0:
        __lowerCAmelCase							     =     [[] for i in range(lowerCAmelCase_					)]
        __lowerCAmelCase							     =     self.root**next_ncol

        # First half of next step
        __lowerCAmelCase							     =     1
        for j in range(self.c_max_length // (next_ncol * 2)					):
           for i in range(lowerCAmelCase_					):
              new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]					)
           current_root *= root
        # Second half of next step
        __lowerCAmelCase							     =     1
        for j in range(self.c_max_length // (next_ncol * 2)					):
           for i in range(lowerCAmelCase_					):
              new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]					)
           current_root *= root
        # Update
        __lowerCAmelCase							     =     new_dft
        __lowerCAmelCase							     =     next_ncol // 2
     return dft[0]






  def     lowercase		(    self			:		Optional[int]					)							->						Any:
     __lowerCAmelCase							     =     self.__dft('A'					)
     __lowerCAmelCase							     =     self.__dft('B'					)
     __lowerCAmelCase							     =     [[dft_a[i] * dft_b[i] for i in range(self.c_max_length					)]]
     del dft_a
     del dft_b

     # Corner Case
     if len(inverce_c[0]					) <= 1:
        return inverce_c[0]
     # Inverse DFT
     __lowerCAmelCase							     =     2
     while next_ncol <= self.c_max_length:
        __lowerCAmelCase							     =     [[] for i in range(lowerCAmelCase_					)]
        __lowerCAmelCase							     =     self.root ** (next_ncol // 2)
        __lowerCAmelCase							     =     1
        # First half of next step
        for j in range(self.c_max_length // next_ncol					):
           for i in range(next_ncol // 2					):
              # Even positions
              new_inverse_c[i].append(
                  (
                      inverce_c[i][j]
                      + inverce_c[i][j + self.c_max_length // next_ncol]
                  )
                  / 2					)
              # Odd positions
              new_inverse_c[i + next_ncol // 2].append(
                  (
                      inverce_c[i][j]
                      - inverce_c[i][j + self.c_max_length // next_ncol]
                  )
                  / (2 * current_root)					)
           current_root *= root
        # Update
        __lowerCAmelCase							     =     new_inverse_c
        next_ncol *= 2
     # Unpack
     __lowerCAmelCase							     =     [round(x[0].real ,  8					) + round(x[0].imag ,  8					) * 1J for x in inverce_c]

     # Remove leading 0's
     while inverce_c[-1] == 0:
        inverce_c.pop()
     return inverce_c






  def __str__(    self			:		Dict					)							->						int:
     __lowerCAmelCase							     =     'A = ' + ' + '.join(
         f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyA[: self.len_A]					)					)
     __lowerCAmelCase							     =     'B = ' + ' + '.join(
         f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyB[: self.len_B]					)					)
     __lowerCAmelCase							     =     'A*B = ' + ' + '.join(
         f"""{coef}*x^{i}""" for coef, i in enumerate(self.product					)					)

     return f"""{a}\n{b}\n{c}"""


# Unit tests
if __name__ == "__main__":
     import doctest

     doctest.testmod()

