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2017 AMC 10B Problem 17

2017 AMC 10B Problem 17AMC 10 Step by Step

Name: Date: September 13, 2026

Basic Counting · difficulty 3 of 5 · about 5 min · position P16-20

Call a positive integer $\textbf{monotonous}$ 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?

A)
$1024$
B)
$1524$
C)
$1533$
D)
$1536$
E)
$2048$

Answer

Key insight

An increasing number is a nonempty subset of {1,...,9} and a decreasing one a subset of {0,...,9} other than {0}; subtract the 9 one-digit numbers counted twice.

Solution

A strictly increasing or strictly decreasing number is determined by its set of digits: sort the set one way or the other. So count digit sets.

Increasing numbers: the digits are distinct and $0$ cannot appear (it would have to come first). Any nonempty subset of $\{1, \dots, 9\}$ works: $2^9 - 1 = 511$ numbers.

Decreasing numbers: any nonempty subset of $\{0, 1, \dots, 9\}$ , written largest to smallest, gives a valid positive integer, except the subset $\{0\}$ alone. That is $2^{10} - 1 - 1 = 1022$ numbers.

Overlap: the one-digit numbers $1$ through $9$ appear in both lists ( $9$ numbers). Nothing else does, since a number with two or more digits cannot be both increasing and decreasing.

Total: $511 + 1022 - 9 = 1524$ .

The answer is $\boxed{\textbf{(B)}\ 1524}$ .

Why this works

The bijection "monotone number $\leftrightarrow$ set of digits" converts an ordering problem into a subset count, where $2^n$ does all the work. The only care needed is at the edges: the digit $0$ is allowed in decreasing numbers but not increasing ones, and one-digit numbers belong to both families.

The trap

Forgetting that 0 may appear as the last digit of a decreasing number (e.g. 987620), or double counting the single-digit numbers.

Common mistakes

Forgetting that 0 may appear as the last digit of a decreasing number (e.g. 987620), or double counting the single-digit numbers.
Excluding one-digit numbers from both counts and then forgetting to add them back, or leaving $\{0\}$ in the decreasing count to get $1525$ .

Techniques

Map the objects to something easier to count · Split into exhaustive cases and handle each