A laptop lock has a 4 digit combination. Each character can be any digit between 0-8. The only restriction is that all 4 characters cannot be the same (e.g. 0000, 1111, 2222,... etc.). How many combinations are possible?

To find the total number of valid combinations for the laptop lock, we need to calculate the following:

1. **Total possible combinations without restriction:**
   - The lock has 4 digits, and each digit can be any number between 0 and 8, so there are 9 possible choices for each digit.
   - The total number of combinations is:
     \[
     9 \times 9 \times 9 \times 9 = 9^4
     \]
   - Calculate \(9^4\):
     \[
     9^4 = 6561
     \]
   So, there are 6,561 possible combinations without any restrictions.

2. **Subtract the invalid combinations (all four digits are the same):**
   - The invalid combinations are those where all four digits are identical (e.g., 0000, 1111, 2222, ..., 8888).
   - There are 9 such combinations (0000, 1111, 2222, ..., 8888), one for each digit.

3. **Calculate the number of valid combinations:**
   - Subtract the 9 invalid combinations from the total number of combinations:
     \[
     6561 - 9 = 6552
     \]

So, there are **6,552** valid combinations for the laptop lock.

Would you like more details or have any questions? Here are some related topics:

1. How would the number of combinations change if there were 5 digits instead of 4?
2. What if the restriction was that no two adjacent digits could be the same?
3. How would the problem change if the digits could be any number from 0 to 9?
4. How do we handle restrictions on combinations in more complex problems?
5. What is the probability of guessing a valid combination on the first try?
6. How does the concept of restrictions affect combinatorial calculations?
7. Can you explain the concept of permutations in this context?
8. How would the solution change if the first digit had more or fewer possible values?

**Tip:** When dealing with restrictions in combinatorial problems, it's often helpful to first calculate the total possibilities without restrictions and then subtract the cases that violate the restrictions.

Learn how to calculate the number of possible combinations for a 4-digit laptop lock using combinatorial principles. Understand the total number of unrestricted combinations and how to subtract invalid combinations to find the valid ones. Suitable for students in Grades 7-9.

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