A family is planning to build a house. They can either build a base model or add any of the following options: Security system, deck, hardwood flooring. How many different variations of the house are possible?

To solve this problem, we need to consider all the different combinations of options that the family can choose for their house. The options are:

1. Security system
2. Deck
3. Hardwood flooring

Each option can either be included or not included, so there are two choices (include or not) for each option.

Since the choices are independent, the total number of variations can be found by multiplying the number of choices for each option.

- For the base model: 1 choice (they always start with the base model).
- For the security system: 2 choices (include or not include).
- For the deck: 2 choices (include or not include).
- For the hardwood flooring: 2 choices (include or not include).

The total number of different variations of the house is calculated as:

\[
\text{Total Variations} = 1 \times 2 \times 2 \times 2 = 8
\]

So, there are 8 different variations of the house possible.

Do you want details on how to visualize these combinations or have any other questions?

Here are 5 related questions:

1. What if the family wanted to add a garage as an additional option?
2. How would the number of variations change if they decided they must include at least one option?
3. Can you list all the 8 possible combinations?
4. How would the number of variations change if some options were dependent on others (e.g., hardwood flooring requires a deck)?
5. What if the family could choose multiple levels of quality for each option (e.g., basic or premium security system)?

**Tip:** When calculating combinations, always check if the choices are independent or if there are any dependencies that might affect the total count.

Learn how to calculate the different variations of a house based on options like security system, deck, and hardwood flooring. Understand the application of combinatorics and the multiplication principle in determining possible combinations. Suitable for students in Grades 6-8.

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