

from math import sqrt



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							int      )			->      bool:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (
				    number >= 0
				), "'number' must been an int and positive"

				A__        =			True

				# 0 and 1 are none primes.
				if number <= 1:
								A__        =			False

				for divisor in range(2	,       int(round(sqrt(SCREAMING_SNAKE_CASE_      )      )      ) + 1      ):
								# if 'number' divisible by 'divisor' then sets 'status'
								# of false and break up the loop.
								if number % divisor == 0:
												A__        =			False
												break

    # precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'status' must been from type bool"

				return status



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Union[str, Any]      )			->      Any:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (n > 2), "'N' must been an int and > 2"

				# beginList: contains all natural numbers from 2 up to N
				A__        =			list(range(2	,       n + 1      )      )

				A__        =			[]  # this list will be returns.

				# actual sieve of erathostenes
				for i in range(len(SCREAMING_SNAKE_CASE_      )      ):
								for j in range(i + 1	,       len(SCREAMING_SNAKE_CASE_      )      ):
												if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
																A__        =			0

    # filters actual prime numbers.
				A__        =			[x for x in begin_list if x != 0]

				# precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'ans' must been from type list"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							List[Any]      )			->      Tuple:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (n > 2), "'N' must been an int and > 2"

				A__        =			[]

				# iterates over all numbers between 2 up to N+1
				# if a number is prime then appends to list 'ans'
				for number in range(2	,       n + 1      ):
								if is_prime(SCREAMING_SNAKE_CASE_      ):
												ans.append(SCREAMING_SNAKE_CASE_      )

    # precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'ans' must been from type list"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							str      )			->      Optional[Any]:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and number >= 0, "'number' must been an int and >= 0"

				A__        =			[]  # this list will be returns of the function.

				# potential prime number factors.

				A__        =			2

				A__        =			number

				if number == 0 or number == 1:
								ans.append(SCREAMING_SNAKE_CASE_      )

				# if 'number' not prime then builds the prime factorization of 'number'
				elif not is_prime(SCREAMING_SNAKE_CASE_      ):
								while quotient != 1:
												if is_prime(SCREAMING_SNAKE_CASE_      ) and (quotient % factor == 0):
																ans.append(SCREAMING_SNAKE_CASE_      )
																quotient /= factor
												else:
																factor += 1

				else:
								ans.append(SCREAMING_SNAKE_CASE_      )

				# precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'ans' must been from type list"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Dict      )			->      int:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (
				    number >= 0
				), "'number' bust been an int and >= 0"

				A__        =			0

				# prime factorization of 'number'
				A__        =			prime_factorization(SCREAMING_SNAKE_CASE_      )

				A__        =			max(SCREAMING_SNAKE_CASE_      )

				# precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'ans' must been from type int"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							str      )			->      Union[str, Any]:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (
				    number >= 0
				), "'number' bust been an int and >= 0"

				A__        =			0

				# prime factorization of 'number'
				A__        =			prime_factorization(SCREAMING_SNAKE_CASE_      )

				A__        =			min(SCREAMING_SNAKE_CASE_      )

				# precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'ans' must been from type int"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							str      )			->      Union[str, Any]:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'number' must been an int"
				assert isinstance(number % 2 == 0	,       SCREAMING_SNAKE_CASE_      ), "compare bust been from type bool"

				return number % 2 == 0



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Optional[Any]      )			->      str:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ), "'number' must been an int"
				assert isinstance(number % 2 != 0	,       SCREAMING_SNAKE_CASE_      ), "compare bust been from type bool"

				return number % 2 != 0



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Tuple      )			->      int:




				'''simple docstring'''





				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (number > 2) and is_even(SCREAMING_SNAKE_CASE_      )
				), "'number' must been an int, even and > 2"

				A__        =			[]  # this list will returned

				# creates a list of prime numbers between 2 up to 'number'
				A__        =			get_prime_numbers(SCREAMING_SNAKE_CASE_      )
				A__        =			len(SCREAMING_SNAKE_CASE_      )

				# run variable for while-loops.
				A__        =			0
				A__        =			None

				# exit variable. for break up the loops
				A__        =			True

				while i < len_pn and loop:
								A__        =			i + 1

								while j < len_pn and loop:
												if prime_numbers[i] + prime_numbers[j] == number:
																A__        =			False
																ans.append(prime_numbers[i]      )
																ans.append(prime_numbers[j]      )

												j += 1

								i += 1

				# precondition
				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and (len(SCREAMING_SNAKE_CASE_      ) == 2)
				    and (ans[0] + ans[1] == number)
				    and is_prime(ans[0]      )
				    and is_prime(ans[1]      )
				), "'ans' must contains two primes. And sum of elements must been eq 'number'"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							List[Any]	,       SCREAMING_SNAKE_CASE_:							Union[str, Any]      )			->      Any:




				'''simple docstring'''





				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and (numbera >= 0)
				    and (numbera >= 0)
				), "'number1' and 'number2' must been positive integer."

				A__        =			0

				while numbera != 0:
								A__        =			numbera % numbera
								A__        =			numbera
								A__        =			rest

				# precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (
				    numbera >= 0
				), "'number' must been from type int and positive"

				return numbera



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Union[str, Any]	,       SCREAMING_SNAKE_CASE_:							List[str]      )			->      Optional[int]:




				'''simple docstring'''





				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and (numbera >= 1)
				    and (numbera >= 1)
				), "'number1' and 'number2' must been positive integer."

				A__        =			1  # actual answer that will be return.

				# for kgV (x,1)
				if numbera > 1 and numbera > 1:
								# builds the prime factorization of 'number1' and 'number2'
								A__        =			prime_factorization(SCREAMING_SNAKE_CASE_      )
								A__        =			prime_factorization(SCREAMING_SNAKE_CASE_      )

				elif numbera == 1 or numbera == 1:
								A__        =			[]
								A__        =			[]
								A__        =			max(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )

				A__        =			0
				A__        =			0

				A__        =			[]  # captured numbers int both 'primeFac1' and 'primeFac2'

				# iterates through primeFac1
				for n in prime_fac_a:
								if n not in done:
												if n in prime_fac_a:
																A__        =			prime_fac_a.count(SCREAMING_SNAKE_CASE_      )
																A__        =			prime_fac_a.count(SCREAMING_SNAKE_CASE_      )

																for _ in range(max(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )      ):
																				ans *= n

												else:
																A__        =			prime_fac_a.count(SCREAMING_SNAKE_CASE_      )

																for _ in range(SCREAMING_SNAKE_CASE_      ):
																				ans *= n

												done.append(SCREAMING_SNAKE_CASE_      )

    # iterates through primeFac2
				for n in prime_fac_a:
								if n not in done:
												A__        =			prime_fac_a.count(SCREAMING_SNAKE_CASE_      )

												for _ in range(SCREAMING_SNAKE_CASE_      ):
																ans *= n

												done.append(SCREAMING_SNAKE_CASE_      )

    # precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (
				    ans >= 0
				), "'ans' must been from type int and positive"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							List[Any]      )			->      Dict:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (n >= 0), "'number' must been a positive int"

				A__        =			0
				A__        =			2  # this variable holds the answer

				while index < n:
								index += 1

								ans += 1  # counts to the next number

								# if ans not prime then
								# runs to the next prime number.
								while not is_prime(SCREAMING_SNAKE_CASE_      ):
												ans += 1

    # precondition
				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and is_prime(
				    SCREAMING_SNAKE_CASE_      ), "'ans' must been a prime number and from type int"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Dict	,       SCREAMING_SNAKE_CASE_:							Any      )			->      Dict:




				'''simple docstring'''





				assert (
				    is_prime(SCREAMING_SNAKE_CASE_      ) and is_prime(SCREAMING_SNAKE_CASE_      ) and (p_number_a < p_number_a)
				), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

				A__        =			p_number_a + 1  # jump to the next number

				A__        =			[]  # this list will be returns.

				# if number is not prime then
				# fetch the next prime number.
				while not is_prime(SCREAMING_SNAKE_CASE_      ):
								number += 1

				while number < p_number_a:
								ans.append(SCREAMING_SNAKE_CASE_      )

								number += 1

								# fetch the next prime number.
								while not is_prime(SCREAMING_SNAKE_CASE_      ):
												number += 1

    # precondition
				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and ans[0] != p_number_a
				    and ans[len(SCREAMING_SNAKE_CASE_      ) - 1] != p_number_a
				), "'ans' must been a list without the arguments"

				# 'ans' contains not 'pNumber1' and 'pNumber2' !
				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							str      )			->      Dict:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (n >= 1), "'n' must been int and >= 1"

				A__        =			[]  # will be returned.

				for divisor in range(1	,       n + 1      ):
								if n % divisor == 0:
												ans.append(SCREAMING_SNAKE_CASE_      )

    # precondition
				assert ans[0] == 1 and ans[len(SCREAMING_SNAKE_CASE_      ) - 1] == n, "Error in function getDivisiors(...)"

				return ans



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							List[Any]      )			->      Optional[int]:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (
				    number > 1
				), "'number' must been an int and >= 1"

				A__        =			get_divisors(SCREAMING_SNAKE_CASE_      )

				# precondition
				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and (divisors[0] == 1)
				    and (divisors[len(SCREAMING_SNAKE_CASE_      ) - 1] == number)
				), "Error in help-function getDivisiors(...)"

				# summed all divisors up to 'number' (exclusive), hence [:-1]
				return sum(divisors[:-1]      ) == number



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							int	,       SCREAMING_SNAKE_CASE_:							Optional[int]      )			->      List[Any]:




				'''simple docstring'''





				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and (denominator != 0)
				), "The arguments must been from type int and 'denominator' != 0"

				# build the greatest common divisor of numerator and denominator.
				A__        =			gcd(abs(SCREAMING_SNAKE_CASE_      )	,       abs(SCREAMING_SNAKE_CASE_      )      )

				# precondition
				assert (
				    isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				    and (numerator % gcd_of_fraction == 0)
				    and (denominator % gcd_of_fraction == 0)
				), "Error in function gcd(...,...)"

				return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							Dict      )			->      Dict:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (n >= 0), "'n' must been a int and >= 0"

				A__        =			1  # this will be return.

				for factor in range(1	,       n + 1      ):
								ans *= factor

				return ans





def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							str      )			->      str:




				'''simple docstring'''





				assert isinstance(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) and (n >= 0), "'n' must been an int and >= 0"

				A__        =			0
				A__        =			1
				A__        =			1  # this will be return

				for _ in range(n - 1      ):
								A__        =			ans
								ans += fiba
								A__        =			tmp

				return ans




import math

lowerCAmelCase__	   = 1_0
lowerCAmelCase__	   = 7
lowerCAmelCase__	   = BALLS_PER_COLOUR * NUM_COLOURS



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							int = 2_0      )			->      str:




				'''simple docstring'''





				A__        =			math.comb(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
				A__        =			math.comb(NUM_BALLS - BALLS_PER_COLOUR	,       SCREAMING_SNAKE_CASE_      )

				A__        =			NUM_COLOURS * (1 - missing_colour / total)

				return F'{result:.9f}'


if __name__ == "__main__":
	print(solution(2_0))

