a safe has a 4 - digit lock code that does not include zero as a digit and no digit is repeated. what is the…

Explanation:

Step1: Determine available digits

Since no - zero and no - repeated digits, available digits are 1 - 9, so we find permutation of 9 things taken 4 at a time. The formula for permutations is $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 9$ and $r=4$. $P(9,4)=\frac{9!}{(9 - 4)!}=\frac{9!}{5!}=9\times8\times7\times6 = 3024$.

Step2: Determine even non - zero digits

Even non - zero digits are 2, 4, 6, 8. We find permutation of 4 things taken 4 at a time. Using the permutation formula $P(n,r)$ with $n = 4$ and $r = 4$, $P(4,4)=\frac{4!}{(4 - 4)!}=\frac{4!}{0!}=4! = 4\times3\times2\times1=24$.

Step3: Calculate probability

Probability $P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Here, number of favorable outcomes is 24 and total number of outcomes is 3024.

Answer:

To find the total number of outcomes for this event, find the permutation of 9 things taken 4 at a time. The total number of outcomes is 3024. The total number of favorable outcomes is a permutation of 4 things taken 4 at a time. The probability that the lock code consists of all even digits is 24 out of 3024.