AMC 10A/12A 2025 (Problem 24/23)
Call a positive integer fair if no digit is used more than once, it has no s, and no digit is adjacent to two greater digits. For example, , , and are fair, but , , and are not fair. How many four-digit positive integers are fair?
Let be the number of digits of the fair number. Base your casework on .
Case 1:
Case 2:
Case 3:
We should select 3 different digits from to , which is .
WLOG, assume we selected and .
We can't place in the middle position since no digit is adjacent to two greater digits.
So, for there are 2 ways to place it, then for and , there are 2 and 1 ways respectively.
We should select 3 different digits from to , which is .
WLOG, assume we selected and .
We can't place in the middle position since no digit is adjacent to two greater digits.
So, for there are 2 ways to place it, then for and , there are 2 and 1 ways respectively.
Case 4: (Similarly)
Case :
Ans
Ans
Ans
Ans
(C)