A briefcase has a three-digit lock code that does not include zero as a digit.

What is the probability that the lock code consists of all even digits if the same digit is not used more than once?

A. 24 out of 504
B. 60 out of 504
C. 24 out of 84
D. 60 out of 84



Answer :

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]