Question

Call a positive integer fair if no digit is used more than once, it has no $0$ s, and no digit is adjacent to two greater digits. For example, $23, 196,$ and $12463$ are fair, but $1546, 320,$ and $34321$ are not fair. How many fair positive integers are there?

$511$

$2{,}584$

$9{,}841$

$17{,}711$

$19{,}682$

Answer: C

Concepts:combinations binomial theorem

Difficulty rating: 2380

Solution:

A fair number's digits must increase up to its largest digit $m$ and then decrease. Indeed, the first ascent after any descent would begin at a digit smaller than both of its neighbors. For $k$ digits, choose the digit set from $1$ to $9$ in $\binom{9}{k}$ ways. Each of the $k-1$ digits below $m$ independently goes on the increasing left side or the decreasing right side, after which its position is forced. Hence the total is $\sum_{k=1}^{9}\binom{9}{k}2^{k-1}$ $= \tfrac12\big((1+2)^9-1\big)$ $= \tfrac{3^9-1}{2}$ $=9841.$ Therefore, the answer is C.

2025 AMC 10A Problem 24

Answer: C

Difficulty rating: 2380

Solution:

Problem 24 in Other Years