
'''simple docstring'''





from typing import Any

import numpy as np


class 			UpperCamelCase__	:






 def __init__(      self	:Optional[int]						,					_A	:np.ndarray						,					_A	:int      )					->      None:







       '''simple docstring'''



       if np.any(tableau[:, -1]						,					where=tableau[:, -1] < 0      ):
             raise ValueError('RHS must be > 0'      )

       __A													=      tableau
       __A						,   __A													=      tableau.shape

       # Number of decision variables x1, x2, x3...
       __A													=      n_vars

       # Number of artificial variables to be minimised
       __A													=      len(np.where(tableau[self.n_vars : -1] == -1      )[0]      )

       # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
       __A													=      (self.n_art_vars > 0) + 1

       # Number of slack variables added to make inequalities into equalities
       __A													=      self.n_rows - self.n_stages

       # Objectives for each stage
       __A													=      ['max']

       # In two stage simplex, first minimise then maximise
       if self.n_art_vars:
             self.objectives.append('min'      )

       __A													=      ['']

       # Index of current pivot row and column
       __A													=      None
       __A													=      None

       # Does objective row only contain (non)-negative values?
       __A													=      False






 @staticmethod
 def 						lowercase_    (      *_A	:int      )					->      list[str]:







       '''simple docstring'''



       if len(_A      ) != 3:
             raise ValueError('Must provide n_vars, n_slack, and n_art_vars'      )

       if not all(x >= 0 and isinstance(_A						,					_A      ) for x in args      ):
             raise ValueError('All arguments must be non-negative integers'      )

       # decision | slack | artificial
       __A													=      ['x', 's', 'a']
       __A													=      []
       for i in range(3      ):
             for j in range(args[i]      ):
                   titles.append(string_starts[i] + str(j + 1      )      )
       titles.append('RHS'      )
       return titles






 def 						lowercase_    (      self	:Optional[int]						,					_A	:np.ndarray      )					->      tuple[Any, Any]:







       '''simple docstring'''



       __A													=      self.objectives[-1]

       # Find entries of highest magnitude in objective rows
       __A													=      (objective == 'min') - (objective == 'max')
       __A													=      np.argmax(sign * tableau[0, : self.n_vars]      )

       # Choice is only valid if below 0 for maximise, and above for minimise
       if sign * self.tableau[0, col_idx] <= 0:
             __A													=      True
             return 0, 0

       # Pivot row is chosen as having the lowest quotient when elements of
       # the pivot column divide the right-hand side

       # Slice excluding the objective rows
       __A													=      slice(self.n_stages						,					self.n_rows      )

       # RHS
       __A													=      tableau[s, -1]

       # Elements of pivot column within slice
       __A													=      tableau[s, col_idx]

       # Array filled with nans
       __A													=      np.full(self.n_rows - self.n_stages						,					np.nan      )

       # If element in pivot column is greater than zeron_stages, return
       # quotient or nan otherwise
       __A													=      np.divide(_A						,					_A						,					out=_A						,					where=divisor > 0      )

       # Arg of minimum quotient excluding the nan values. n_stages is added
       # to compensate for earlier exclusion of objective columns
       __A													=      np.nanargmin(_A      ) + self.n_stages
       return row_idx, col_idx






 def 						lowercase_    (      self	:str						,					_A	:np.ndarray						,					_A	:int						,					_A	:int      )					->      np.ndarray:







       '''simple docstring'''



       __A													=      tableau[row_idx].copy()

       __A													=      piv_row[col_idx]

       # Entry becomes 1
       piv_row *= 1 / piv_val

       # Variable in pivot column becomes basic, ie the only non-zero entry
       for idx, coeff in enumerate(tableau[:, col_idx]      ):
             tableau[idx] += -coeff * piv_row
       __A													=      piv_row
       return tableau






 def 						lowercase_    (      self	:Union[str, Any]						,					_A	:np.ndarray      )					->      np.ndarray:







       '''simple docstring'''



       self.objectives.pop()

       if not self.objectives:
             return tableau

       # Slice containing ids for artificial columns
       __A													=      slice(-self.n_art_vars - 1						,					-1      )

       # Delete the artificial variable columns
       __A													=      np.delete(_A						,					_A						,					axis=1      )

       # Delete the objective row of the first stage
       __A													=      np.delete(_A						,					0						,					axis=0      )

       __A													=      1
       self.n_rows -= 1
       __A													=      0
       __A													=      False
       return tableau






 def 						lowercase_    (      self	:List[Any]      )					->      dict[Any, Any]:







       '''simple docstring'''



       for _ in range(100      ):
             # Completion of each stage removes an objective. If both stages
             # are complete, then no objectives are left
             if not self.objectives:
                   __A													=      self.generate_col_titles(
                       self.n_vars						,					self.n_slack						,					self.n_art_vars      )

                   # Find the values of each variable at optimal solution
                   return self.interpret_tableau(self.tableau						,					self.col_titles      )

             __A						,   __A													=      self.find_pivot(self.tableau      )

             # If there are no more negative values in objective row
             if self.stop_iter:
                   # Delete artificial variable columns and rows. Update attributes
                   __A													=      self.change_stage(self.tableau      )
             else:
                   __A													=      self.pivot(self.tableau						,					_A						,					_A      )
       return {}






 def 						lowercase_    (      self	:List[str]						,					_A	:np.ndarray						,					_A	:list[str]      )					->      dict[str, float]:







       '''simple docstring'''



       __A													=      {'P': abs(tableau[0, -1]      )}

       for i in range(self.n_vars      ):
             # Gives ids of nonzero entries in the ith column
             __A													=      np.nonzero(tableau[:, i]      )
             __A													=      len(nonzero[0]      )

             # First entry in the nonzero ids
             __A													=      nonzero[0][0]
             __A													=      tableau[nonzero_rowidx, i]

             # If there is only one nonzero value in column, which is one
             if n_nonzero == nonzero_val == 1:
                   __A													=      tableau[nonzero_rowidx, -1]
                   __A													=      rhs_val

        # Check for basic variables
       for title in col_titles:
             # Don't add RHS or slack variables to output dict
             if title[0] not in "R-s-a":
                   output_dict.setdefault(_A						,					0      )
       return output_dict


if __name__ == "__main__":
 import doctest

 doctest.testmod()









from math import factorial




def        __UpperCamelCase     (      _A					:							Union[str, Any]      ,  _A					:							Tuple		)				->Any:



							"""simple docstring"""
							return factorial(_A		) / (factorial(_A		) * factorial(n - r		))




def        __UpperCamelCase     (      )				->Any:



							"""simple docstring"""
							lowerCamelCase_								=0

							for i in range(1      ,  101		):
														for j in range(1      ,  i + 1		):
																					if combinations(_A      ,  _A		) > 1E6:
																												total += 1
							return total


if __name__ == "__main__":
	print(solution())
