Seating with a ladies-only seat among five seats: Two men (distinct) and one woman board a bus with 5 vacant seats, one seat reserved for ladies only. The woman may or may not take the reserved seat (men cannot). In how many ways can these three passengers occupy seats?

Introduction / Context:We count placements of three distinct passengers (M1, M2, W) into five seats where one is ladies-only (L). Men cannot occupy L. The woman can sit on L or any normal seat, and if she avoids L, that seat remains empty. Use clean casework by whether W uses L.

Given Data / Assumptions:

• Seats: L (ladies-only) + 4 normal seats.
• M1, M2 cannot sit in L; W may or may not sit in L.
• Each person takes exactly one seat; two seats remain empty.

Concept / Approach:

• Case 1: W sits in L; men fill 2 of 4 normal seats in order.
• Case 2: W sits in a normal seat; men fill 2 of the remaining 3 normal seats; L must stay empty.

Step-by-Step Solution:

Verification / Alternative check:Counting by first selecting seats then permuting people yields the same result; constraints force L to be taken by W or left empty.

Why Other Options Are Wrong:

• 48 and 60 overcount by allowing men on L or by double-counting orders.
• 15 is far too small.

Common Pitfalls:

• Accidentally placing a man in the ladies-only seat.

Final Answer:36