A brand of automobile comes in five different styles, with four types of engines, with two types

of transmissions, and in eight colors.

**a** How many autos would a dealer have to stock if he included one for each style–engine–

transmission combination?

**b** How many would a distribution center have to carry if all colors of cars were stocked for

each combination in part (a)?

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.

A brand of automobile comes in five different styles, with four types of engines, with two types of transmissions, and in eight colors.

(a) How many autos would a dealer have to stock if he included one for each style–engine–transmission combination?

(b) How many would a distribution center have to carry if all colors of cars were stocked for each combination in part (a)?

Learn how to calculate the total number of automobile combinations with different styles, engines, transmissions, and colors. This combinatorics problem involves multiplying choices for each category and is suitable for students in Grades 6-8. Step-by-step explanation provided.

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