Difference between revisions of "2017 AMC 10B Problems/Problem 17"

Revision as of 18:46, 16 February 2017

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

Solution: Answer Choices

by e_power_pi_times_i

It adds in $9$ at the end, but we know since it is MAA, it is probably a troll question, so we look at the answers. $(C)$ looks likely, as it is just $(B)+9$ , but we remember that MAA is trolly so it is probably $\boxed{\textbf{(B) }1524}$ .

Legit 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 $2^{10}+2^9 - 9 - 1 - 1 - 1 = \boxed{\textbf{(B) }1524}$ .

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 <math>\textbf{(A)}\ 1024\qquad\textbf{(B)}\ 1524\qquad\textbf{(C)}\ 1533\qquad\textbf{(D)}\ 1536\qquad\textbf{(E)}\ 2048</math>
-==Solution: Answer Choices==
+==Solution==
+===Solution: Answer Choices===
 by e_power_pi_times_i

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