2017 AMC 10B Problem 17

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

Answer: B

Concepts:subsets digits bijection

Difficulty rating: 1660

Small hint:

Increasing numbers are determined by their digit set

Big hint:

Decreasing numbers may include

0

, but the number

0

itself is not positive

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.