Question

若一个正整数是一位数，或者它的数字从左到右读时构成严格递增或严格递减的序列，则称它为 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：子集数字双射

难度评级：1660

解答：

严格递增的正整数对应于 $\{1,2,3,4,5,6,7,8,9\}$ 的非空子集，把选出的数字按递增顺序写出。因此有 $2^9-1=511$ 个。

严格递减的正整数对应于 $\{0,1,2,3,4,5,6,7,8,9\}$ 的子集，把选出的数字按递减顺序写出，但要排除空集和 $\{0\}$ 。因此有 $2^{10}-2=1022$ 个。

一位数 $1$ 到 $9$ 被两边都算了一次，所以总数为 $511+1022-9=1524$ 。所以正确答案是 B。

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 第 17 题

答案：B

难度评级：1660

解答：

其他年份的第 17 题