On Monday, $6$ students went to the tutoring center at the same time, and each one was randomly assigned to one of the $6$ tutors on duty. On Tuesday, the same $6$ students showed up, the same $6$ tutors were on duty, and the students were again randomly assigned to the tutors. What is the probability that exactly $2$ students met with the same tutor both Monday and Tuesday?
$$\textbf{(A) } \frac{1}{16} \qquad\textbf{(B) } \frac{3}{16} \qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{1}{2}$$
Difficulty: 36/100 - Relatively straightforward derangement or casework problem.
Core Concepts: casework, derangements, probability
Challenges: Contestants who don't know the derangements formula may struggle finding the number of ways exactly two people get the same tutor.
Techniques: Recognizing that once two students are paired with the same tutors, the derangements formula may be applied.
Error-prone Steps: Messing up the number of ways to derange the other four students.
Ideal Time:
Experienced: ≤ 1 min
Intermediate: ≤ 2-3 min
Beginner: ≤ 6 min