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

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 $\ge 2,$ giving $D = 2^{10} - 1 - 10 = 1013.$ So $|U - D| = |502 - 1013| = 511.$ Therefore, the answer is E.

2023 AMC 10B 第 16 题

答案：E

难度评级：1800

解答：

其他年份的第 16 题