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How many ways can you pick the cars so not all are the same size?

For this problem, assume that you are to pick 3 cars from the motor pool, which contains 8 subcompact cars, 8 compact cars, and 5 midsize cars.

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Jon P. | Knowledgeable Math, Science, SAT, ACT tutor - Harvard honors gradKnowledgeable Math, Science, SAT, ACT tu...
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How many ways can you pick 3 cars, forgetting about size?  There are 21 cars, so there are 21! / (3! (21 - 3)!) ways to pick 3.  That's 21! / (3!/18!) = 1330.
 
Now, in how many of those selections would they all be the same size?  There are 3 ways you could do that:
 
1. Pick 3 subcompacts.  There are 8! / (3! (8-3))! = 8! / (3!5!) = 56 way to do that.
2. Pick 3 compacts.  There are the same number of compacts as subcompacts, so there are 56 ways to do this too.
3. Pick 3 midsize cares.  There are 5! / (3! (5-3))! = 5! / (3!2!) = 10.
 
So there are a total of 56+56+10 = 122 ways to pick 3 cars that are all the same.
 
So subtract that from the number of ways to pick 3 cars (1330) to get the number of ways to pick 3 cars that are NOT all the same.  That's 1208.