AMC 12A 2025 (Problem 18)
How many ordered triples of different positive integers less than or equal to satisfy , , and ?
WLOG, assume .
We will multiply by at the end to account for all possible orderings.
Since and , we should just focus on .
We will multiply by at the end to account for all possible orderings.
Since and , we should just focus on .
Case 1:
No valid cases.
No valid cases.
Case 2:
Then , 1 case.
Then , 1 case.
Case 3:
cases.
cases.
Case 4:
cases.
cases.
Case 5:
Bad cases:
cases.
Bad cases:
cases.
Case 6:
Bad cases:
cases.
Bad cases:
cases.
(C)