a certain train has 10 cars that are being lined up on a track. one of the cars is the engine, and another is the caboose. the engine will be the first car in line. the caboose will be the last car in line. in how many ways can the cars be lined up?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Permutations", "Fundamental Counting Principle" ], "new_concepts": [], "current_concepts": [ "Permutations", "Fundamental Counting Principle" ] } ## Identify fixed positions $$ \begin{aligned} &\text{Total cars} = 10\ &\text{Position 1 (First)} = \text{Engine (1 choice)}\ &\text{Position 10 (Last)} = \text{Caboose (1 choice)} \end{aligned} $$ ## Calculate remaining arrangements $$ \begin{aligned} &\text{Remaining cars} = 10 - 2 = 8\ &\text{Remaining positions} = 8\ &\text{Number of ways} = 8! \end{aligned} $$ ## Compute final value $$ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 $$ A certain train has 10 cars that are being lined up on a track. One of the cars is the engine, and another is the caboose. The engine will be the first car in line. The caboose will be the last car in line. In how many ways can the cars be lined up? 40320 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Permutations" ] }

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Explanation:

Identify fixed positions

Calculate remaining arrangements

Compute final value

Answer: