AMC12 2017 B

AMC12 2017 B · Q11

AMC12 2017 B · Q11. It mainly tests Basic counting (rules of product/sum), 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的序列。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

Answer (B): 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 $2^9-1=511$. 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 $\emptyset$ and $\{0\}$. The number of these is $2^{10}-2=1022$. The single-digit numbers are included in both sets, so there are $511+1022-9=1524$ monotonous positive integers.

答案（B）：只有一位数或数字递增的单调正整数，可以与集合 $\{1,2,3,4,5,6,7,8,9\}$ 的非空子集建立一一对应。这样的子集个数为 $2^9-1=511$。只有一位数或数字递减的单调正整数，可以与集合 $\{0,1,2,3,4,5,6,7,8,9\}$ 中除 $\emptyset$ 和 $\{0\}$ 之外的子集建立一一对应。这样的子集个数为 $2^{10}-2=1022$。一位数同时包含在两类中，因此单调正整数共有 $511+1022-9=1524$ 个。

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