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: Calculate number of all - even - digit lock codes

The non - zero even digits are 2, 4, 6, 8. The number of ways to arrange 4 non - repeated non - zero even digits is given by the permutation formula (P(n,r)=\frac{n!}{(n - r)!}), where (n = 4) (the number of non - zero even digits) and (r=4) (the number of digits in the code). So (P(4,4)=\frac{4!}{(4 - 4)!}=4!=24).

Step2: Calculate number of lock codes without all even digits

The total number of 4 - digit lock codes with non - zero non - repeated digits is (9\times8\times7\times6 = 3024) (9 choices for the first digit, 8 for the second, 7 for the third, and 6 for the fourth). The number of lock codes without all even digits is (3024-24 = 3000).

Step3: Calculate the probability

The probability (P) of a 4 - digit lock code without all even digits is (\frac{\text{number of lock codes without all even digits}}{\text{total number of lock codes}}). So (P=\frac{3000}{3024}=\frac{125}{126}).

Answer:

There are 24 ways to get a lock code with all even digits. There are 3000 ways to get a lock code without all even digits. The probability of a 4 - digit lock code without all even digits is (\frac{125}{126}).