To determine the probability that a 4-digit lock code consists of all even digits, we first need to identify the constraints provided in the problem.
Step 1: Total Outcomes
Since the lock code cannot include the digit zero and no digit can be repeated, the valid digits we can use are {1, 2, 3, 4, 5, 6, 7, 8, 9}. This gives us a total of 9 digits to choose from.
The number of different 4-digit lock codes can be calculated by finding the number of permutations of 9 digits taken 4 at a time. This can be calculated using the formula for permutations:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
where \( n \) is the total number of options and \( r \) is the number of slots to fill. In this case:
\[ P(9, 4) = \frac{9!}{(9-4)!} = \frac{9!}{5!} = 9 \times 8 \times 7 \times 6 = 3024 \]
Step 2: Favorable Outcomes
Next, we determine the number of favorable outcomes where all four digits are even. The even digits from the possible set {1, 2, 3, 4, 5, 6, 7, 8, 9} are {2, 4, 6, 8}. This gives us 4 even digits to choose from.
We want the number of ways to create a 4-digit code using these 4 digits, where all digits must be different. The number of ways to arrange these 4 digits is given by the number of permutations of 4 digits taken 4 at a time:
\[ P(4, 4) = \frac{4!}{(4-4)!} = 4! = 24 \]
Step 3: Probability Calculation
Now, we can calculate the probability that a 4-digit lock code consists of all even digits by dividing the number of favorable outcomes by the total outcomes:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{24}{3024} \]
To simplify:
\[ \frac{24}{3024} = \frac{1}{126} \]
Conclusion
Therefore, the probability that the lock code consists of all even digits is
\[ \frac{1}{126} \text{ out of } 3024. \]
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