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 solve this problem, we need to consider the cases where the three cars selected are of different sizes or two cars are of the same size and the third car is of a different size.
Case 1: Three cars of different sizes
To calculate the number of ways to choose three cars of different sizes, we can multiply the number of choices for each size category.
Number of choices for subcompact cars (6): We need to choose 1 car from the 6 available, which can be done in 6 ways.
Number of choices for compact cars (6): We need to choose 1 car from the 6 available, which can be done in 6 ways.
Number of choices for midsize cars (5): We need to choose 1 car from the 5 available, which can be done in 5 ways.
So the total number of ways to pick 3 cars of different sizes is 6 * 6 * 5 = 180 ways.
Case 2: Two cars of the same size and one car of a different size
To calculate the number of ways to choose two cars of the same size and one car of a different size, we need to consider each size category.
Number of choices for subcompact cars (6): We need to choose 2 cars from the 6 available, which can be done in C(6, 2) ways (6 choose 2, which is calculated as 6! / (2! * (6-2)!)). This equals 15 ways.
Number of choices for compact cars (6): We need to choose 2 cars from the 6 available, which can be done in 15 ways (same as above).
Number of choices for midsize cars (5): We need to choose 1 car from the 5 available, which can be done in 5 ways.
So the total number of ways to pick 2 cars of the same size and 1 car of a different size is 15 * 15 * 5 = 1125 ways.
Therefore, the total number of ways to pick three cars so that not all are the same size is 180 + 1125 = 1305 ways.