1. **Stating the problem:** We need to find the number of different 5-digit combinations possible on a lock with 5 wheels labeled 0 to 9 under different conditions. 2. **Understanding the problem:** Each wheel can show digits from 0 to 9, so there are 10 possible digits per wheel. 3. **Case a: No digit is repeated.** - For the first wheel, there are 10 choices. - For the second wheel, since no repetition is allowed, there are 9 choices left. - For the third wheel, 8 choices remain. - For the fourth wheel, 7 choices remain. - For the fifth wheel, 6 choices remain. 4. **Formula used:** The number of permutations of 10 digits taken 5 at a time is given by: $$P(10,5) = 10 \times 9 \times 8 \times 7 \times 6$$ 5. **Calculating:** $$10 \times 9 = 90$$ $$90 \times 8 = 720$$ $$720 \times 7 = 5040$$ $$5040 \times 6 = 30240$$ 6. **Answer:** The number of different 5-digit combinations possible if no digit is repeated is **30240**.