





"""simple docstring"""



def 		__lowercase					(       snake_case_  :  int						)       ->bool:



    '''simple docstring'''







    __A   :						Union[str, Any]									=					(1 + 24 * n) ** 0.5
    return ((1 + root) / 6) % 1 == 0







def 		__lowercase					(       snake_case_  :  int = 5000						)       ->int:



    '''simple docstring'''







    __A   :						str									=					[(i * (3 * i - 1)) // 2 for i in range(1    ,snake_case_						)]
    for i, pentagonal_i in enumerate(snake_case_						):
        for j in range(snake_case_    ,len(snake_case_						)						):
            __A   :						int									=					pentagonal_nums[j]
            __A   :						str									=					pentagonal_i + pentagonal_j
            __A   :						int									=					pentagonal_j - pentagonal_i
            if is_pentagonal(snake_case_						) and is_pentagonal(snake_case_						):
                return b

    return -1


if __name__ == "__main__":
       print(f'''{solution() = }''')











"""simple docstring"""



from math import factorial







def 		__lowercase					(       snake_case_  :  int    ,snake_case_  :  int						)       ->int:



    '''simple docstring'''







    if n < k or k < 0:
        raise ValueError('''Please enter positive integers for n and k where n >= k'''						)
    return factorial(snake_case_						) // (factorial(snake_case_						) * factorial(n - k						))


if __name__ == "__main__":
       print(
           """The number of five-card hands possible from a standard""",
           f'''fifty-two card deck is: {combinations(52, 5)}\n''',
       )

       print(
           """If a class of 40 students must be arranged into groups of""",
           f'''4 for group projects, there are {combinations(40, 4)} ways''',
           """to arrange them.\n""",
       )

       print(
           """If 10 teams are competing in a Formula One race, there""",
           f'''are {combinations(10, 3)} ways that first, second and''',
           """third place can be awarded.""",
       )




