A certain train has 11 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?
Question
A certain train has 11 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?
Solution
The problem can be solved by using the concept of permutations.
Here's how:
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We know that the engine will be the first car and the caboose will be the last car. So, we are left with 9 cars (11 total - 1 engine - 1 caboose = 9 cars) that can be arranged in any order.
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The number of ways to arrange these 9 cars is given by the formula for permutations, which is n!, where n is the number of items to arrange. The "!" symbol denotes factorial, which means to multiply the number by every positive integer less than it. For example, 5! = 54321 = 120.
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So, the number of ways to arrange the 9 cars is 9!, which equals 362,880.
Therefore, there are 362,880 ways to line up the cars on the track.
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