Notice that some of the cars are identical. We must take this into account when calculating the number of ways we can arrange them.
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Practice makes perfect
To determine how many ways we can arrange the cars, we first have to determine how many cars there are in total.
r|l
First class & 2 cars
Second class & 5 cars
Restaurant & 1 car
Total & 8 cars
We have a group of 8 cars and we want to know how many ways they can be arranged. Notice that this is a permutation where two types of cars — the first class and second class — occur multiple times. Therefore, we want to use the following formula.
n!/r_1!r_2!...
In this formula n is the number of items and r_1, r_2,... is the number of times that item 1, item 2, and so on repeats. As already explained there are 8 cars, which means n= 8. Also, two car-types occur twice and five times, respectively. This means we have r_1= 2. and r_2= 5.
8!/2! 5!
Let's calculate this on a graphing calculator.
