A customer forgets a four digit code for an Automatic Teler Machine (ATM) in a bank. However, he remembers that this code consists of digits 2, 3, 6 and 9. Find the largest possible number of trials necessary to obtain the correct code.

Answer

To find the largest possible number of trials necessary to obtain the correct four-digit code for the ATM, we need to determine how many different arrangements can be made using the digits 2, 3, 6, and 9. Since the code consists of these four distinct digits, we can use the concept of permutations.

### Step-by-Step Solution:

1. **Identify the Digits**: The digits available for the ATM code are 2, 3, 6, and 9.

2. **Count the Digits**: There are a total of 4 distinct digits.

3. **Calculate the Number of Arrangements**: Since all digits must be used and they are distinct, we can find the total number of arrangements (permutations) of these 4 digits. The formula for the number of permutations of n distinct objects is given by n! (n factorial).

\[ \text{Number of arrangements} = 4! = 4 \times 3 \times 2 \times 1 \]

4. **Perform the Calculation**:
\[ 4! = 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 1 = 24 \]

5. **Conclusion**: Therefore, the largest possible number of trials necessary to obtain the correct code is **24**.