How do I calculate the number of combinations from something like a 3-digit code and 6 buttons with turn on or off.

To calculate the number of combinations for a situation where you have a 3-digit code and 6 buttons that can each be turned on or off, you need to consider both aspects separately and then combine the results.

### Step 1: Calculate the number of combinations for the 3-digit code
- A 3-digit code usually consists of digits from 0 to 9.
- So, each digit has 10 possible choices.
- For a 3-digit code, the number of possible combinations is:

\[
10 \times 10 \times 10 = 10^3 = 1000
\]

### Step 2: Calculate the number of combinations for the 6 buttons
- Each button has 2 states: on or off.
- Therefore, for 6 buttons, the number of possible combinations is:

\[
2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 = 64
\]

### Step 3: Combine the results
- Since the 3-digit code and the 6-button configurations are independent of each other, the total number of combinations is the product of the two:

\[
1000 \times 64 = 64000
\]

So, there are **64,000 possible combinations** in total.

Would you like more details, or do you have any other questions?

Here are 5 related questions you might be interested in:

1. How would the calculation change if the 3-digit code allowed repeated digits?
2. What if there were 8 buttons instead of 6?
3. How can the number of combinations be affected if the buttons have more than two states (e.g., on, off, and standby)?
4. What is the number of combinations if the code were 4 digits instead of 3?
5. How would you calculate the number of combinations if there were restrictions, like specific digits or buttons that must be used?

**Tip:** When dealing with combinations, always ensure to check if there are any restrictions or conditions that might reduce the total count.

Learn how to calculate the total number of combinations for a system with a 3-digit code and 6 buttons that can be turned on or off. This detailed solution includes key concepts in combinatorics and basic arithmetic. Perfect for students in Grades 7-9.

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