For this problem, assume that you are to pick 3 cars from the motor pool, which contains 6 subcompact cars, 6 compact cars, and 5 midsize cars.
How many ways can you pick the cars so not all are the same size?
To find the number of ways to pick 3 cars from the motor pool so that not all of them are the same size, we can use the principle of complementary counting.
First, let's calculate the total number of ways to pick 3 cars from the motor pool without any restrictions. This can be done using combinations:
6 subcompact cars choose 3 = C(6, 3) = 20
6 compact cars choose 3 = C(6, 3) = 20
5 midsize cars choose 3 = C(5, 3) = 10
Next, let's calculate the number of ways to pick 3 cars where all of them are of the same size. We can do this by finding the sum of selecting 3 subcompact cars, 3 compact cars, and 3 midsize cars:
1 way to select 3 subcompact cars
1 way to select 3 compact cars
1 way to select 3 midsize cars
Therefore, the total number of ways to pick 3 cars where all of them are the same size is 1 + 1 + 1 = 3.
Finally, we can subtract the number of ways to pick 3 cars where all of them are the same size from the total number of ways to pick 3 cars without restrictions:
Total number of ways without restrictions - Number of ways with all cars of the same size = 20 + 20 + 10 - 3 = 47
Hence, there are 47 ways to select 3 cars from the motor pool so that not all of them are the same size.
So what you want is one car from each of the groups, clearly the number of ways is
6x6x5 or 180 ways