To understand the number of possible combinations for a 5-wheel combination lock with digits from 0 to 9, we should consider the different scenarios based on the repetition of digits.
a) If no digit is repeated:
When no digit is repeated, we need to choose 5 different digits from the set of 10 numbers (0-9). The first wheel can have any of the 10 digits. The second wheel can have any of the remaining 9 digits, the third wheel can choose from 8 digits, the fourth from 7, and the last from 6 digits. This gives us:
Combinations = 10 × 9 × 8 × 7 × 6 = 30,240
b) If digits are repeated:
If digits can be repeated, each of the 5 wheels can independently choose from any of the 10 digits, allowing for the same digit to appear multiple times. Hence, for each wheel, there are 10 choices. This results in:
Combinations = 10 × 10 × 10 × 10 × 10 = 100,000
c) If successive digits cannot be the same:
In this scenario, the first wheel can still choose any of the 10 digits. However, for each subsequent wheel, it cannot choose the digit that was just picked. Therefore, the first wheel has 10 options, while each of the next four wheels has 9 options. The calculation is as follows:
Combinations = 10 × 9 × 9 × 9 × 9 = 73,440
In summary, we have:
- No repetition: 30,240 combinations
- With repetition: 100,000 combinations
- No successive digits: 73,440 combinations