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?
$\textbf{(A) } 176 \qquad\textbf{(B) } 192 \qquad\textbf{(C) } 432 \qquad\textbf{(D) } 464 \qquad\textbf{(E) } 608$
Difficulty: 35/100 — Somewhat tricky casework; easy to miss a case or compute a case incorrectly.
Core Concepts: casework, constructive counting
Challenges: Know how to encounter the situation of what happens when there is a $2$ in the passcode. Being careful about which integers to include and which integers to not include in the code.
Techniques: Constructive counting; building the passcode one digit at a time.
Error-prone Steps: Entirely forgetting a case or computing the number of passcodes for each case incorrectly.
Ideal Time:
Experienced: ≤ 1.5 min
Intermediate: ≤ 2.5 min
Beginner: ≤ 5 min