Question

using permutations in calculating theoretical probability
a safe has a 4 - digit lock code that does not include zero as a digit and no digit is repeated. what is the probability that the lock code consists of all even digits?
to find the total number of outcomes for this event, find the permutation of things taken 4 at a time.
the total number of outcomes is.
the total number of favorable outcomes is a permutation of things taken 4 at a time.
the probability that the lock code consists of all even digits is out of 3,024.

Explanation:

Step1: Determine available digits for total outcomes

Since no - zero and no - repeated digits, there are 9 non - zero digits (1 - 9). We are forming a 4 - digit code, so we find the 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 available digits for favorable outcomes

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

Step3: Calculate the probability

The probability $P$ is the number of favorable outcomes divided by the number of total outcomes. So $P=\frac{24}{3024}=\frac{1}{126}$.

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.