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. How many fair positive integers are there?

$511$

$2{,}584$

$9{,}841$

$17{,}711$

$19{,}682$

Solution:

In a fair number the digits climb up to the largest digit $m$ and then fall; otherwise some digit would be trapped between two bigger ones. So count by size. For $k$ digits, pick the digit set from $1$ to $9$ in $\binom{9}{k}$ ways. The largest is $m,$ and each of the remaining $k - 1$ digits chooses to sit left or right of $m$ ( $2^{k-1}$ ways), which pins down the number. Sum over $k$ : $\sum_{k=1}^{9}\binom{9}{k}2^{k-1} = \tfrac12\big((1 + 2)^9 - 1\big) = \tfrac{3^9 - 1}{2} = 9841.$ Therefore, the answer is C.

2025 AMC 10A Problem 24

Solution:

Problem 24 in Other Years