2017 AMC 10B Problem 17

Problem 17 of 25IntermediateNumber Theory Counting & Probability

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?

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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.

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