2023 AMC 10B 第 16 题

Question

定义“升数”为一个至少

2

位的正整数，其数字从左到右严格递增。类似地，定义“降数”为一个至少

2

位的正整数，其数字从左到右严格递减。例如，

258

是升数，

8620

是降数。令

U

为所有升数的总数，令

D

为所有降数的总数。求

|U - D|

。

Define an upno to be a positive integer of

2

or more digits where the digits are strictly increasing moving left to right. Similarly, define a downno to be a positive integer of

2

or more digits where the digits are strictly decreasing moving left to right. For instance, the number

258

is an upno and

8620

is a downno. Let

U

equal the total number of upnos and let

D

equal the total number of downnos. What is

|U - D|?

512

10

0

9

511

答案：E

知识点：子集数字

难度评级：1800

小提示：

每个这样的数只是选择一组数字，再按规定顺序写出。

Each such number is just a choice of its set of digits, written in the required order

大提示：

递增数使用

\{1, \ldots, 9\}

中的数字；递减数还可以包含末尾的

0

。

Increasing numbers use digits from

\{1, \ldots, 9\};

decreasing numbers may also include a trailing

0

解答：

一个“升数”就是选择至少

2

个数字并按递增顺序写出。数字

0

不能出现，因为它不能在首位，也不能跟在更小的数字后面。因此数字来自

\{1, \ldots, 9\}

，所以

U = 2^9 - 1 - 9 = 502

。一个“降数”可以以

0

结尾，所以可从

\{0, \ldots, 9\}

中选择

\ge 2

个数字，得到

D = 2^{10} - 1 - 10 = 1013

。因此

|U - D| = |502 - 1013| = 511

。所以正确答案是 E。

An upno is just a choice of at least

2

digits, written in increasing order. A

0

can never appear: it can’t lead and can’t follow a smaller digit. So the digits come from

\{1, \ldots, 9\},

giving

U = 2^9 - 1 - 9 = 502.

A downno can end in

0,

so its digits are any subset of

\{0, \ldots, 9\}

of size

\ge 2,