In the previous question, if the lock opens in only one arrangement of 4 digits, how many unsuccessful attempts are possible in which lock can not be opened?

Options

A. `4^(9)-1`
B. `9^(4 )-1`
C. `""^(9)C_4`
D. none of these

Answer

To solve the problem of how many unsuccessful attempts are possible for a lock that opens with only one specific arrangement of four digits, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Digits**:
We are given that there are 9 non-zero digits (1 through 9).

2. **Calculate the Total Possible Arrangements**:
Since the lock has 4 rings, and each ring can be filled with any of the 9 digits, the total number of arrangements (or combinations) of digits for the lock can be calculated using the formula for permutations with repetition:
\[ \text{Total arrangements} = 9^4 \]
This is because each of the 4 positions can be filled in 9 different ways.

3. **Calculate the Total Arrangements**:
\[ 9^4 = 6561 \]

4. **Identify the Successful Attempt**:
The problem states that there is only one arrangement that successfully opens the lock.

5. **Calculate the Number of Unsuccessful Attempts**:
To find the number of unsuccessful attempts, we subtract the successful attempt from the total arrangements:
\[ \text{Unsuccessful attempts} = \text{Total arrangements} - \text{Successful attempts} \]
\[ \text{Unsuccessful attempts} = 6561 - 1 = 6560 \]

### Final Answer:
The number of unsuccessful attempts possible in which the lock cannot be opened is **6560**.

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