Question

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 $|U - D|?$

$512$

$10$

$0$

$9$

$511$

Answer: E

Concepts:subsets digits

Difficulty rating: 1800

Small Hint:

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

Big Hint:

Increasing numbers use digits from $\{1, \ldots, 9\};$ decreasing numbers may also include a trailing $0$

Solution:

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 Problem 16

Answer: E

Difficulty rating: 1800

Small Hint:

Big Hint:

Solution:

Problem 16 in Other Years