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

Question

For this problem, assume that you are to pick 4 cars from the motor pool, which contains 7 subcompact cars, 6 compact cars, and 4 midsize cars. 

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

Answer 1

Establish the total number of possible draws of 4 cars from 17 cars:

₁₇C₄ or 17! ÷ 4! ÷ (17 − 4)! gives (17 × 16 × 15 × 14) ÷ 24 or 2380.

Next, there are ₇C₄ or 7! ÷4! ÷ (7 − 4)! or 35 possible outcomes that

will produce 4 cars that are all sub-compact.

Also, there are ₆C₄ or 6! ÷4! ÷ (6 − 4)! or 15 possible outcomes that

will produce 4 cars that are all compact.

Then there is ₄C₄ or 4! ÷ 4! ÷ (4 − 4)! or 1 possible draw that will give

4 cars that are all mid-size.

Finally take the number of 4-car-draws that will produce

4 cars (that are not all the same size) out of 17 total cars  

as 2380 − 35 − 15 − 1 or 2329.