2017 AMC 10B Problems/Problem 17

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The following problem is from both the 2017 AMC 12B #11 and 2017 AMC 10B #17, so both problems redirect to this page.

Problem

Call a positive integer $monotonous$ if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?

$\textbf{(A)}\ 1024\qquad\textbf{(B)}\ 1524\qquad\textbf{(C)}\ 1533\qquad\textbf{(D)}\ 1536\qquad\textbf{(E)}\ 2048$

Solution

The number of one-digit numbers that work is $\binom{10}{1}$, and the number of two-digit integers that work is $\binom{10}{2} + \binom{9}2$. We use similar logic for three-digit integers, four digit integers, etc. Summing, we have $2^{10}+2^9 - 9 - 1 - 1$, and we need to subtract another 1 for the 0 case, so the answer is $2^{10}+2^9 - 9 - 1 - 1 - 1 = \boxed{\textbf{(B) }1524}$.

See Also

2017 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AMC 10 Problems and Solutions

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