Question

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

Digits are 1 - 9, so there are 9 non - zero digits.

Step2: Calculate total number of outcomes

The formula for permutations is $P(n,r)=\frac{n!}{(n - r)!}$. Here $n = 9$ and $r=4$, so $P(9,4)=\frac{9!}{(9 - 4)!}=\frac{9!}{5!}=9\times8\times7\times6 = 3024$.

Step3: Determine even non - zero digits

Even non - zero digits are 2, 4, 6, 8, so $n = 4$.

Step4: Calculate favorable outcomes

Using permutation formula with $n = 4$ and $r = 4$, $P(4,4)=\frac{4!}{(4 - 4)!}=\frac{4!}{0!}=4\times3\times2\times1=24$.

Step5: Calculate probability

Probability $P=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{24}{3024}=\frac{1}{126}$.

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 3,024.