




from __future__ import annotations





def        __UpperCAmelCase						(							lowerCamelCase_ :     str  ,							lowerCamelCase_ :     list[str] | None = None  ,							lowerCamelCase_ :     dict[str, float] | None = None  ,							lowerCamelCase_ :     bool = False  ,							)       ->  tuple[int, float, str]:







			"""simple docstring"""





			SCREAMING_SNAKE_CASE_   : str					  =      cipher_alphabet or [chr(lowerCamelCase_  ) for i in range(97  ,							1_23  )]

			# If the argument is None or the user provided an empty dictionary
			if not frequencies_dict:
						# Frequencies of letters in the english language (how much they show up)
						SCREAMING_SNAKE_CASE_   : Union[str, Any]					  =      {
						    'a': 0.0_8_4_9_7,
						    'b': 0.0_1_4_9_2,
						    'c': 0.0_2_2_0_2,
						    'd': 0.0_4_2_5_3,
						    'e': 0.1_1_1_6_2,
						    'f': 0.0_2_2_2_8,
						    'g': 0.0_2_0_1_5,
						    'h': 0.0_6_0_9_4,
						    'i': 0.0_7_5_4_6,
						    'j': 0.0_0_1_5_3,
						    'k': 0.0_1_2_9_2,
						    'l': 0.0_4_0_2_5,
						    'm': 0.0_2_4_0_6,
						    'n': 0.0_6_7_4_9,
						    'o': 0.0_7_5_0_7,
						    'p': 0.0_1_9_2_9,
						    'q': 0.0_0_0_9_5,
						    'r': 0.0_7_5_8_7,
						    's': 0.0_6_3_2_7,
						    't': 0.0_9_3_5_6,
						    'u': 0.0_2_7_5_8,
						    'v': 0.0_0_9_7_8,
						    'w': 0.0_2_5_6_0,
						    'x': 0.0_0_1_5_0,
						    'y': 0.0_1_9_9_4,
						    'z': 0.0_0_0_7_7,
						}
			else:
						# Custom frequencies dictionary
						SCREAMING_SNAKE_CASE_   : List[Any]					  =      frequencies_dict

			if not case_sensitive:
						SCREAMING_SNAKE_CASE_   : int					  =      ciphertext.lower()

			# Chi squared statistic values
			SCREAMING_SNAKE_CASE_   : dict[int, tuple[float, str]]					  =      {}

			# cycle through all of the shifts
			for shift in range(len(lowerCamelCase_  )  ):
						SCREAMING_SNAKE_CASE_   : List[str]					  =      ''

						# decrypt the message with the shift
						for letter in ciphertext:
									try:
												# Try to index the letter in the alphabet
												SCREAMING_SNAKE_CASE_   : List[Any]					  =      (alphabet_letters.index(letter.lower()  ) - shift) % len(
												    lowerCamelCase_  )
												decrypted_with_shift += (
												    alphabet_letters[new_key].upper()
												    if case_sensitive and letter.isupper()
												    else alphabet_letters[new_key]
												)
									except ValueError:
												# Append the character if it isn't in the alphabet
												decrypted_with_shift += letter

						SCREAMING_SNAKE_CASE_   : int					  =      0.0

						# Loop through each letter in the decoded message with the shift
						for letter in decrypted_with_shift:
									if case_sensitive:
												SCREAMING_SNAKE_CASE_   : Tuple					  =      letter.lower()
												if letter in frequencies:
															# Get the amount of times the letter occurs in the message
															SCREAMING_SNAKE_CASE_   : Any					  =      decrypted_with_shift.lower().count(lowerCamelCase_  )

															# Get the excepcted amount of times the letter should appear based
															# on letter frequencies
															SCREAMING_SNAKE_CASE_   : List[str]					  =      frequencies[letter] * occurrences

															# Complete the chi squared statistic formula
															SCREAMING_SNAKE_CASE_   : Union[str, Any]					  =      ((occurrences - expected) ** 2) / expected

															# Add the margin of error to the total chi squared statistic
															chi_squared_statistic += chi_letter_value
									else:
												if letter.lower() in frequencies:
															# Get the amount of times the letter occurs in the message
															SCREAMING_SNAKE_CASE_   : str					  =      decrypted_with_shift.count(lowerCamelCase_  )

															# Get the excepcted amount of times the letter should appear based
															# on letter frequencies
															SCREAMING_SNAKE_CASE_   : List[str]					  =      frequencies[letter] * occurrences

															# Complete the chi squared statistic formula
															SCREAMING_SNAKE_CASE_   : str					  =      ((occurrences - expected) ** 2) / expected

															# Add the margin of error to the total chi squared statistic
															chi_squared_statistic += chi_letter_value

        # Add the data to the chi_squared_statistic_values dictionary
						SCREAMING_SNAKE_CASE_   : Optional[Any]					  =      (
						    chi_squared_statistic,
						    decrypted_with_shift,
						)

			# Get the most likely cipher by finding the cipher with the smallest chi squared
			# statistic
			def chi_squared_statistic_values_sorting_key(lowerCamelCase_ :     int  ) -> tuple[float, str]:
						return chi_squared_statistic_values[key]

			SCREAMING_SNAKE_CASE_   : int					  =      min(
			    lowerCamelCase_  ,							key=lowerCamelCase_  ,							)

			# Get all the data from the most likely cipher (key, decoded message)
			(
			    (
			    SCREAMING_SNAKE_CASE_
			)  ,    (
			    SCREAMING_SNAKE_CASE_
			)  ,    
			)   : Tuple					  =      chi_squared_statistic_values[most_likely_cipher]

			# Return the data on the most likely shift
			return (
			    most_likely_cipher,
			    most_likely_cipher_chi_squared_value,
			    decoded_most_likely_cipher,
			)









def        __UpperCAmelCase						(							lowerCamelCase_ :     List[str]  ,							lowerCamelCase_ :     Optional[Any]  )       ->  int:







			"""simple docstring"""





			SCREAMING_SNAKE_CASE_   : List[Any]					  =      [0 for i in range(r + 1  )]
			# nc0 = 1
			SCREAMING_SNAKE_CASE_   : Dict					  =      1
			for i in range(1  ,							n + 1  ):
						# to compute current row from previous row.
						SCREAMING_SNAKE_CASE_   : Optional[int]					  =      min(lowerCamelCase_  ,							lowerCamelCase_  )
						while j > 0:
									c[j] += c[j - 1]
									j -= 1
			return c[r]




print(binomial_coefficient(n=10, r=5))



