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Counting principle involving a specified arrangement89°FA certain train has 9 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. Thecaboose will be the last car in line. In how many ways can the cars be lined up?0ExplanationI need help with this math problem.

Sagot :

Given:

[tex]\begin{gathered} Total(cars)=9 \\ Engine(car)=1 \\ Caboose(car)=1 \end{gathered}[/tex]

To Determine: The many ways the cars can be lined up

Solution

Since we have 9 cars, the first must be engine car. The number of ways of that is 1!

The last must be caboose, then the number of ways of that is 1!.

In between will be remaining 7 ways to arrange the ways. This can be done in 7! ways

Therefore, the number of ways to arrange the 9 cars would be

[tex]\begin{gathered} Ways=1!\times7!\times1! \\ Ways=1\times7\times6\times5\times4\times3\times2\times1\times1 \\ Ways=5040 \end{gathered}[/tex]

Hence, the number of ways the cars can be lined up is 5040 ways

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