2025 AMC 10A Problem 12

Problem 12 of 25IntermediateCounting & Probability

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?

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Solution

No digit is

0,

so digits run from

1

9.

Put the single even digit in the first slot for now and multiply by

4

at the end to place it. Split on that even digit. If it’s the prime

2,

then the three odd digits all have to be non-prime, so each is

1

9,

giving

2^3 = 8

ways. Otherwise the even digit is

4, 6,

8

(

3

choices), and exactly one of the odd digits is prime, worth

3, 5,

7

(

3

choices) in one of the

3

odd positions, while the other two odds come from

\{1, 9\}

(

2^2

ways). That’s

3 \cdot 3 \cdot 3 \cdot 4 = 108.

Altogether,

4(8 + 108) = 464.

Therefore, the answer is D.