AMC 10A 2025 (Problem 12)
CombinatoricsMedium
Carlos uses a 4-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is 0. How many 4-digit passcodes satisfy these conditions?
Base your casework on whether the even digit is prime or not.
Case 1: (eooo)
There are four places to position . The other odd digits shouldn't be prime, so they can only be or .
There are four places to position . The other odd digits shouldn't be prime, so they can only be or .
Case 2:
There are 3 choices for the even number and 4 ways to place it. We must select one odd prime , so there are 3 choices for that as well, and 3 possible places to put it.
The other 2 digits can only be or :
There are 3 choices for the even number and 4 ways to place it. We must select one odd prime , so there are 3 choices for that as well, and 3 possible places to put it.
The other 2 digits can only be or :
(D)
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