There are 3 different cars available to transport 3 girls and 5 boys

The kids can be distributed among the 3 cars:

3 3 and 2

Since the question asks for arrangements with 2 or 3 girls in a car, we'll be adding those two scenarios.

Let's start with 2 girls in a car and no boys.

Two girls can be selected;

3!/2! = 3 ways

A car can be selected 3 ways.

So 3*3 = 9 ways

Now we have 6 people remaining to be distributed among 2 cars.

6!/3!3! = 20 are the ways to select 3 people from 6. This counts the leftover 3 as well, which isn't double counting since the 2 vehicles are different.

So 9*20 = 180

Now let's try 2 girls and 1 boy in a car.

Two girls can be selected 3 ways as above. 1 boy can be selected 5 ways and 1 car can be selected 3 ways, a total of

45 ways

The car with 2 people can be selected 2 ways. The 2 people to go in the car can be selected

5!/2!3! = 10 ways for a total of 10*2 = 20 ways.

Total ways: 45*20 = 900

Finally, let's put 3 girls in 1 car.

3 girls can be selected 1 way. A car can be selected 3 ways, for a total of

3 ways

The car with 2 people can be selected 2 ways. 2 people can be selected from 5

5!/2!3! = 10 ways

So a total of 3*2*10 = 60 ways


Totaling all of these:

180+900+60 = 1140

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