Question

若一个正整数是一位数，或者从左到右读它的数字时，其数字构成严格递增或严格递减的序列，就称它为单调数。例如， $3$ 、 $23578$ 和 $987620$ 是单调数，但 $88$ 、 $7434$ 和 $23557$ 不是。共有多少个单调正整数？

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?

$1024$

$1524$

$1533$

$1536$

$2048$

答案：B

知识点：子集双射容斥原理

难度评级：1590

解答：

严格递增的单调数对应 $\{1, \ldots, 9\}$ 的非空子集，得到 $2^9 - 1 = 511$ 。严格递减的数对应 $\{0, 1, \ldots, 9\}$ 的子集，但要排除 $\varnothing$ 和 $\{0\}$ （不允许前导 $0$ ），得到 $2^{10} - 2 = 1022$ 。九个一位数在两类中被重复计算，所以总数为 $511 + 1022 - 9 = 1524$ 。

所以正确答案是 B。

Strictly increasing monotonous numbers correspond to nonempty subsets of $\{1, \ldots, 9\},$ giving $2^9 - 1 = 511.$ Strictly decreasing ones correspond to subsets of $\{0, 1, \ldots, 9\}$ other than $\varnothing$ and $\{0\}$ (a leading $0$ is not allowed), giving $2^{10} - 2 = 1022.$ The nine single-digit numbers are counted in both, so the total is $511 + 1022 - 9 = 1524.$

Thus, the correct answer is B.

2017 AMC 12B 第 11 题

答案：B

难度评级：1590

解答：

其他年份的第 11 题