Add solution for Project Euler problem 113 (#3109)

* Added solution for Project Euler problem 113. #2695

* Updated formatting and doctests. Reference: #3256
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"""
Project Euler Problem 113: https://projecteuler.net/problem=113
Working from left-to-right if no digit is exceeded by the digit to its left it is
called an increasing number; for example, 134468.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing
number; for example, 66420.
We shall call a positive integer that is neither increasing nor decreasing a
"bouncy" number; for example, 155349.
As n increases, the proportion of bouncy numbers below n increases such that there
are only 12951 numbers below one-million that are not bouncy and only 277032
non-bouncy numbers below 10^10.
How many numbers below a googol (10^100) are not bouncy?
"""
def choose(n: int, r: int) -> int:
"""
Calculate the binomial coefficient c(n,r) using the multiplicative formula.
>>> choose(4,2)
6
>>> choose(5,3)
10
>>> choose(20,6)
38760
"""
ret = 1.0
for i in range(1, r + 1):
ret *= (n + 1 - i) / i
return round(ret)
def non_bouncy_exact(n: int) -> int:
"""
Calculate the number of non-bouncy numbers with at most n digits.
>>> non_bouncy_exact(1)
9
>>> non_bouncy_exact(6)
7998
>>> non_bouncy_exact(10)
136126
"""
return choose(8 + n, n) + choose(9 + n, n) - 10
def non_bouncy_upto(n: int) -> int:
"""
Calculate the number of non-bouncy numbers with at most n digits.
>>> non_bouncy_upto(1)
9
>>> non_bouncy_upto(6)
12951
>>> non_bouncy_upto(10)
277032
"""
return sum(non_bouncy_exact(i) for i in range(1, n + 1))
def solution(num_digits: int = 100) -> int:
"""
Caclulate the number of non-bouncy numbers less than a googol.
>>> solution(6)
12951
>>> solution(10)
277032
"""
return non_bouncy_upto(num_digits)
if __name__ == "__main__":
print(f"{solution() = }")