AMC10 2017 B

AMC10 2017 B · Q17

AMC10 2017 B · Q17. It mainly tests Combinations, Casework.

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?

称一个正整数为单调的，如果它是一位数，或者其数字从左到右阅读时形成严格递增或严格递减序列。例如，3、23578和987620是单调的，但88、7434和23557不是。有多少个单调正整数？

(A) 1024 1024

(B) 1524 1524

(D) 1536 1536

(E) 2048 2048

Answer

Correct choice: (B)

正确答案：（B）

Solution

The monotonous positive integers with one digit or increasing digits can be put into a one-to-one correspondence with the nonempty subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9}. The number of such subsets is 2⁹ − 1 = 511. The monotonous positive integers with one digit or decreasing digits can be put into a one-to-one correspondence with the subsets of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} other than ∅ and {0}. The number of these is 2¹⁰ − 2 = 1022. The single-digit numbers are included in both sets, so there are 511 + 1022 − 9 = 1524 monotonous positive integers.

一位数或递增数字的单调正整数可与{1,2,3,4,5,6,7,8,9}的非空子集一一对应，这样的子集数为2^9−1=511。一位数或递减数字的单调正整数可与{0,1,2,3,4,5,6,7,8,9}的子集（除去∅和{0}）一一对应，这样的子集数为2^10−2=1022。单数字被包含在两个集合中，因此总单调正整数数为511+1022−9=1524。

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