Let's solve each part step by step:

Part (a)

We are given that there are:

• 5 different styles of automobiles,
• 4 types of engines,
• 2 types of transmissions.

We need to find out how many different combinations there are if we include one for each style-engine-transmission combination. To calculate this, we multiply the number of choices for each category:

$\text{Total combinations} = \text{Styles} \times \text{Engines} \times \text{Transmissions}$ $\text{Total combinations} = 5 \times 4 \times 2 = 40$

Thus, the dealer would have to stock 40 cars to cover all style-engine-transmission combinations.

Part (b)

Now, the question asks how many cars a distribution center would have to carry if all colors are stocked for each combination from part (a). We are given that there are 8 colors. So, for each of the 40 combinations calculated in part (a), there are 8 possible colors.

To calculate the total number of autos, we multiply the 40 combinations by the number of colors:

$\text{Total cars} = 40 \times 8 = 320$

Thus, the distribution center would have to carry 320 cars to stock all colors for each style-engine-transmission combination.

Final Answer:

• (a) The dealer would need to stock 40 autos.
• (b) The distribution center would need to carry 320 autos.

Would you like a detailed explanation of any part of this solution?

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Further related questions:

1. What if there were 6 types of engines? How would the answer change for part (a)?
2. How many cars would the distribution center carry if the number of colors increased to 10?
3. If a dealer only stocked 3 styles, how many cars would be required for part (a)?
4. What if each combination had 3 different tire options? How would this affect part (a)?
5. How many total combinations are there if the number of transmission types doubled?

Tip: When solving counting problems like this, remember to multiply the number of choices in each category to find the total number of possible combinations.