The problem asks us to list all possible 3-digit codes where the first digit can be either '$' or 'K', the second digit can be either '4' or '8', and the third digit can be '1', '3', or '5'. We must also verify that the total number of codes we find matches the number predicted by the multiplication principle.

1. Problem Description

The problem asks us to list all possible 3-digit codes where the first digit can be either '$' or 'K', the second digit can be either '4' or '8', and the third digit can be '1', '3', or '5'. We must also verify that the total number of codes we find matches the number predicted by the multiplication principle.

2. Solution Steps

First, we list all the possible combinations:

* Start with '$' as the first digit.

* '$41$'

* '$43$'

* '$45$'

* '$81$'

* '$83$'

* '$85$'

* Start with 'K' as the first digit.

* 'K41'

* 'K43'

* 'K45'

* 'K81'

* 'K83'

* 'K85'

Therefore, the total number of codes is

1

2.

The multiplication principle states that if there are $n_1$ ways to do the first task, $n_2$ ways to do the second task, and $n_3$ ways to do the third task, then the total number of ways to do all three tasks is $n_1 \times n_2 \times n_3$.

In this case, there are 2 choices for the first digit, 2 choices for the second digit, and 3 choices for the third digit.

Total number of codes $= 2 \times 2 \times 3 = 12$

This agrees with the number of codes we listed.

3. Final Answer

The list of all possible 3-digit codes is: $41,$43, $45,$81, $83,$85, K41, K43, K45, K81, K83, K

8

5. The total number of codes is 12, which agrees with the multiplication principle ($2 \times 2 \times 3 = 12$).