Emil is shopping for a convertible. He plans to test drive three cars, a Toyota, a Honda, and a Mazda. He also wants to test drive both the 5-speed standard and the automatic transmission in each, but the Mazda isn't made with an automatic transmission. He would like to test drive the cars both with the trunks empty and with the trunks fully loaded to see how they perform, but the Toyota doesn't have a trunk. How many test drives he will have to do to check all the combination of features?

I came up with 29 because with the Honda you can test all 4 things and then you can arrange those 4 things in a different order to get 16. Then with a Toyota you can only test 2 thigs and then arrange those 2 things in a different order to get 4. Finally, with the Mazda you can test 3 things and then you can arrange those 3 things in different orders to get 9.

I come up with 8.

Toyota: 2 options: 5-speed or automatic (both with no trunk)
Mazda: 2 options:
5-speed w/trunk loaded;
5-speed w/trunk empty
Honda: 4 options:
5-speed w/trunk loaded
5-speed w/trunk empty
automatic w/trunk loaded
automatic w/trunk empty

I lost you when you mentioned "all four things" and "in different orders"

The order of what you test is no important.

To find the number of test drives Emil needs to do to check all the combinations, you can break down the problem into three steps:

Step 1: Calculate the number of test drives for each car.
- For the Honda, Emil can choose between two transmissions (5-speed standard or automatic) and two trunk states (empty or fully loaded). So the total number of test drives for the Honda is 2 (transmissions) x 2 (trunk states) = 4.
- For the Toyota, Emil can choose between two transmissions (5-speed standard or automatic). The Toyota does not have a trunk, so the trunk state is not applicable. Therefore, the total number of test drives for the Toyota is 2.
- For the Mazda, Emil can only choose between one transmission (5-speed standard) because the Mazda does not come with an automatic transmission. Emil can choose between two trunk states (empty or fully loaded). Therefore, the total number of test drives for the Mazda is 1 (transmission) x 2 (trunk states) = 2.

Step 2: Calculate the number of combinations for each car.
- For the Honda, with 4 different features (2 transmissions and 2 trunk states), the number of combinations is 2^4 = 16. This is because each feature has 2 options, so you multiply the number of options for each feature together.
- For the Toyota, with 2 different features (2 transmissions), the number of combinations is 2^2 = 4.
- For the Mazda, with 3 different features (1 transmission and 2 trunk states), the number of combinations is 2^3 = 8.

Step 3: Calculate the total number of test drives.
- To find the total number of test drives, you need to sum up the test drives for each car. So the total number of test drives is 4 (Honda) + 2 (Toyota) + 2 (Mazda) = 8 + 2 + 2 = 12.

Therefore, Emil needs to do a total of 12 test drives to check all the combinations of features.