A number lock on a suitcase has three wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.

Explanation

To solve the problem of how many sequences can be formed with a number lock that has three wheels, each labeled with digits from 0 to 9, and where no digit can repeat, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Digits Available**:
There are 10 digits available (0 to 9).

2. **Determine the Choices for Each Wheel**:
- For the **first wheel**, you can choose any of the 10 digits. So, there are **10 choices**.
- For the **second wheel**, you cannot repeat the digit chosen for the first wheel. Therefore, you have **9 remaining choices**.
- For the **third wheel**, you cannot repeat the digits chosen for the first and second wheels. Thus, you have **8 remaining choices**.

3. **Calculate the Total Number of Sequences**:
The total number of sequences can be calculated by multiplying the number of choices for each wheel:
\[ \text{Total sequences} = 10 \times 9 \times 8 \]
\[ = 720 \]

4. **Determine the Number of Unsuccessful Attempts**:
Since there is only one correct sequence, the number of unsuccessful attempts can be calculated by subtracting the one successful attempt from the total sequences:
\[ \text{Unsuccessful attempts} = \text{Total sequences} - 1 \]
\[ = 720 - 1 = 719 \]

### Final Answers:
- The total number of possible sequences is **720**.
- The number of unsuccessful attempts to open the lock is **719**.