Problem 16 · 2003 AMC 8
Hard
Counting & Probability
careful-counting
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has four seats: one driver's seat, one front passenger seat, and two back passenger seats. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
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Answer: D — 12 arrangements.
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Hint 1 of 2
Handle the seat with a rule attached FIRST — the driver's seat — before the free-for-all of the other seats.
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Hint 2 of 2
Once the driver is locked in, the remaining three people just fill three seats with no restrictions.
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Approach: fill the restricted seat first (the constrained-choice rule)
- The driver's seat is the only one with a rule: just Bonnie or Carlo can sit there, so 2 choices. Settling the restriction first keeps it from tangling the rest.
- With the driver seated, the other 3 people drop into the 3 remaining seats freely: 3! = 3 × 2 × 1 = 6 ways.
- Multiply the independent stages: 2 × 6 = 12 arrangements.
- Worth keeping: in counting problems, fill the most restricted slot first — if you save it for last, the count for the "free" slots changes depending on earlier picks and the multiplication breaks down.
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