Question

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?

$1024$

$1524$

$1533$

$1536$

$2048$

Solution:

The strictly increasing positive integers correspond to the nonempty subsets of $\{1,2,3,4,5,6,7,8,9\}$ , written in increasing order. There are $2^9-1=511$ of these.

The strictly decreasing positive integers correspond to subsets of $\{0,1,2,3,4,5,6,7,8,9\}$ , written in decreasing order, except for the empty set and $\{0\}$ . There are $2^{10}-2=1022$ of these.

The one-digit numbers $1$ through $9$ were counted in both groups, so the total is $511+1022-9=1524$ . Thus, B is the correct answer.

2017 AMC 10B Problem 17

Solution:

Problem 17 in Other Years