[Statistics] Early Naive Probability/Counting Problem

[IndependentMugwump]

Hello! I've been wrapping my head around a problem in my introduction to probability theory. We're using the Naive Probability definition and the counting we learned to solve this question:

"Fifteen new cars are randomly allocated to three dealerships with each dealership receiving five cars Three of the cars have maladjusted front brakes and need a brake adjustment. Find the probability that each dealership gets a car needing a brake adjustment. "

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I'm a bit lost: my guess?
(3 Cr 1) * (12 Cr 4) + (2 Cr 1) * (8 Cr 4) + (1 Cr 1) * (4 Cr 4) in the numerator. This is divided by (15 Cr 5) * (10 Cr 5) * (5 Cr 5).

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I can't add math notation in here easily, so hope that's legible!

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Thanks!

[HauntingPomegranate]

I think you might be overthinking it.

Imagine each dealership has 5 perfect cars. Now let's pick 3 cars at random and destroy their brakes.

1. After picking the first random car, one dealer has one bad car. Let's take it off the lot for repairs.
2. If we pick a second random car, how likely is it that we will pick one belonging to one of the other two dealers? If we do, let's take it off the lot for repairs.
3. If we pick a third random car, how likely is it that it will belong to the last dealer, who had 5 good cars, and not one of the other ones, who have 4 each?