Add solution for Project Euler problem 113 (TheAlgorithms#3109)

* Added solution for Project Euler problem 113. TheAlgorithms#2695

* Updated formatting and doctests. Reference: TheAlgorithms#3256

Author: fpringle

Diff for: project_euler/problem_113/__init__.py

@@ -0,0 +1,75 @@
+"""
+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() = }")