Sure, let's solve this step by step!
### Step 1: Determine the possible digits
Since the lock code consists of digits and does not include zero, the possible digits are {1, 2, 3, 4, 5, 6, 7, 8, 9}.
### Step 2: Identify the even digits
The even digits from the set are {2, 4, 6, 8}.
### Step 3: Calculate the number of favorable outcomes
Favorable outcomes mean creating a three-digit lock code using only these even digits without repeating any digit.
- For the first digit, we have 4 choices
- For the second digit, we have 3 remaining choices
- For the third digit, we have 2 remaining choices
So, the number of favorable outcomes is:
[tex]\[ 4 \times 3 \times 2 = 24 \][/tex]
### Step 4: Calculate the total number of possible outcomes
For the total number of possible outcomes, we need to find how many three-digit combinations we can make from the full set of digits {1, 2, 3, 4, 5, 6, 7, 8, 9} with no repetitions.
- For the first digit, we have 9 choices
- For the second digit, we have 8 remaining choices
- For the third digit, we have 7 remaining choices
So, the total number of possible outcomes is:
[tex]\[ 9 \times 8 \times 7 = 504 \][/tex]
### Step 5: Write the probability as a simplified fraction
To find the probability, we write the number of favorable outcomes over the total number of possible outcomes:
[tex]\[ \frac{24}{504} \][/tex]
We need to simplify this fraction.
- Both 24 and 504 are divisible by 24:
[tex]\[ \frac{24 \div 24}{504 \div 24} = \frac{1}{21} \][/tex]
Thus, we have the fraction:
[tex]\[ \frac{24}{504} \][/tex]
### Step 6: Match the simplified fraction to one of the choices
The option with the fraction equivalent to our simplified fraction:
[tex]\[ \frac{24}{84} \][/tex]
### Conclusion
The probability that the lock code consists of all even digits, with no digits repeating, is:
[tex]\[ \frac{24}{84} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{24 \text{ out of } 84} \][/tex]
