

"""simple docstring"""




import numpy as np
from numpy import floataa
from numpy.typing import NDArray
def 					lowerCAmelCase  (   __UpperCamelCase							,      __UpperCamelCase ):
 '''simple docstring'''





 UpperCAmelCase__				, UpperCAmelCase__   :					Any   				=   np.shape(__UpperCamelCase )

 UpperCAmelCase__   :					NDArray[floataa]   				=   np.zeros((rows, 1)							,      dtype=__UpperCamelCase )
 for row in reversed(range(__UpperCamelCase ) ):
  UpperCAmelCase__   :					List[Any]   				=   0
  for col in range(row + 1							,      __UpperCamelCase ):
   total += coefficients[row, col] * x[col]

  UpperCAmelCase__   :					str   				=   (vector[row] - total) / coefficients[row, row]

 return x
def 					lowerCAmelCase  (   __UpperCamelCase							,      __UpperCamelCase ):
 '''simple docstring'''





 UpperCAmelCase__				, UpperCAmelCase__   :					Dict   				=   np.shape(__UpperCamelCase )
 if rows != columns:
  return np.array(()							,      dtype=__UpperCamelCase )

 # augmented matrix
 UpperCAmelCase__   :					NDArray[floataa]   				=   np.concatenate((coefficients, vector)							,      axis=1 )
 UpperCAmelCase__   :					Dict   				=   augmented_mat.astype("""float64""" )

 # scale the matrix leaving it triangular
 for row in range(rows - 1 ):
  UpperCAmelCase__   :					Dict   				=   augmented_mat[row, row]
  for col in range(row + 1							,      __UpperCamelCase ):
   UpperCAmelCase__   :					List[Any]   				=   augmented_mat[col, row] / pivot
   augmented_mat[col, :] -= factor * augmented_mat[row, :]

 UpperCAmelCase__   :					Optional[Any]   				=   retroactive_resolution(
     augmented_mat[:, 0:columns]							,      augmented_mat[:, columns : columns + 1] )

 return x


if __name__ == "__main__":
      import doctest

      doctest.testmod()








"""simple docstring"""




from math import factorial
def 					lowerCAmelCase  (   __UpperCamelCase							,      __UpperCamelCase ):
 '''simple docstring'''





 return factorial(__UpperCamelCase ) / (factorial(__UpperCamelCase ) * factorial(n - r ))
def 					lowerCAmelCase  (   ):
 '''simple docstring'''





 UpperCAmelCase__   :					Any   				=   0

 for i in range(1							,      101 ):
  for j in range(1							,      i + 1 ):
   if combinations(__UpperCamelCase							,      __UpperCamelCase ) > 1e6:
    total += 1
 return total


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





