Problem Description

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 10¹⁰.

How many numbers below a googol (10¹⁰⁰) are not bouncy?

Analysis

Whenever a problem asks for the number of elements in a set, as does this one, then the solution in typically a counting problem solved with combinometrics.

Our solution uses the general formula that counts the monotonically increasing/decreasing digits ignoring leading zeros.

Solution

Runs < 1 second in Python.

from Euler import binomial
 
n=100
print binomial(n+10,10) + binomial(n+9,9) - 10*n - 2

Comments

More information on the Euler module can be found on the tools page.