https://file.aiquickdraw.com/mathbot/file/17285333251024th1f4ft.jpg

Let's break down and solve the questions presented in the image regarding the combination lock:

### Given:
- The lock has 4 wheels, each labeled with the digits 0 through 9, so there are 10 possible digits (0-9) for each wheel.

#### (a) The number of different 4-digit combinations possible if no digit is repeated.

When no digit is repeated, the choices for each digit reduce as we select the digits. The calculation will be:

- For the 1st digit: 10 possible choices (0-9)
- For the 2nd digit: 9 choices (since one digit is already used)
- For the 3rd digit: 8 choices
- For the 4th digit: 7 choices

So, the total number of combinations is:

\[
10 \times 9 \times 8 \times 7 = 5040
\]

#### (b) The number of different 4-digit combinations possible if digits can be repeated.

If digits can be repeated, then for each of the four digits, we have 10 possible choices (0-9).

- For the 1st digit: 10 choices
- For the 2nd digit: 10 choices
- For the 3rd digit: 10 choices
- For the 4th digit: 10 choices

The total number of combinations is:

\[
10 \times 10 \times 10 \times 10 = 10,000
\]

#### (c) The number of different 4-digit combinations possible if successive digits must be different.

Here, the first digit can be any of the 10 digits, but each subsequent digit must differ from the one immediately preceding it:

- For the 1st digit: 10 possible choices
- For the 2nd digit: 9 choices (it must be different from the 1st)
- For the 3rd digit: 9 choices (it must be different from the 2nd)
- For the 4th digit: 9 choices (it must be different from the 3rd)

Thus, the total number of combinations is:

\[
10 \times 9 \times 9 \times 9 = 7290
\]

### Final Answers:
- **(a)** 5040
- **(b)** 10,000
- **(c)** 7290

Let me know if you want details or further clarification.

---

Here are some related questions to expand your understanding:

1. How would the number of combinations change if the lock had 5 wheels instead of 4?
2. What if the digits ranged from 1 to 9 instead of 0 to 9?
3. Can you calculate how many combinations there are if only odd digits are allowed?
4. How would the number of combinations change if at least one digit must be repeated?
5. What is the probability of guessing the correct combination on the first try?

**Tip:** When considering "no repetition" scenarios, the number of choices decreases as you use more digits, which is important in combinatorial problems.

A small combination lock on a suitcase has 4 wheels, each labeled with the 10 digits 0 to 9. How many 4-digit combinations are possible if no digit is repeated? If digits can be repeated? If successive digits must be different?

Explore how to calculate the number of possible 4-digit combinations for a suitcase lock, considering different conditions such as no repetition, allowing repetition, and requiring successive digits to be different. Learn the formulas and principles behind these combinatorics problems.

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