Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
$\textbf{(A) } \frac 16 \qquad \textbf{(B) } \frac 15 \qquad \textbf{(C) } \frac 29 \qquad \textbf{(D) } \frac 3{13} \qquad \textbf{(E) } \frac 14$
Difficulty: 34/100 — Typical combinatorics problem with many ways to approach; not particularly hard and feasible for beginners.
Core Concepts: casework, constructive counting, arrangements in a circle
Challenges: Determining how to distinguish amongst the students and teachers. Circular permutations may be harder to work with.
Techniques: Multiplying the probability of independent events, simulating the seating of the students and teachers.
Error-prone Steps: Improperly applying combinations and permutations, incorrectly dealing with circular seatings and symmetry.
Ideal Time:
Experienced: ≤ 1 min
Intermediate: ≤ 2-3 min
Beginner: ≤ 4-5 min