Hi...I am working on the problem below and am getting a different answer from th

Question

Hi...I am working on the problem below and am getting a different answer from the solutions manual. The solutions manual shows there are 6 combinations of models you can choose. The problem says "The model has three options, each one of which you can either take or not take." To me that says you have 8 options like I've written out below. Can you help me understand this?

Problem: You want to buy a new car, and you know the model you want. The model has three options, each one of which you can either take or not take, and you have a choice of four colors. So far 100,000 cars of this model have been sold. What is the largest number of cars that you can guarantee to have the same color and the same options as each other?

Solutions manual solution:

1. option 1, option 2, option 3

2. option 1, option 2, option 2

3. option 2, option 1, option 3

4. option 2, option 3, option 1

5. option 3, option 1, option 2

6. option 3, option 2, option 1

My solution:

Considering you have a choice of three options each one you can take or not take, this is how I'm looking at it. The first column represents option 1, second option 2, and third option 2:

1. Y Y Y

2. Y Y N

3. Y N Y

4. N Y Y

5. Y N N

6. N Y N

7. N N Y

8. N N N

Explanation / Answer

Because if you can either take or not take 3 options, that results in 6 possible combinations of the 3 options. You multiply 6 by 4 (number of colors) to get how many different color/option combinations you can have, which is 24.

Then all you do is divide 100,000 by 24. = 4166

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