We are asked to list all possible 2-digit codes where the first digit is one of the letters X, Y, or Z, and the second digit is one of the numbers 5, 6, 7, 8, or 9. Then we must verify that the total number of codes is the same as the number predicted by the multiplication principle.

1. Problem Description

We are asked to list all possible 2-digit codes where the first digit is one of the letters X, Y, or Z, and the second digit is one of the numbers 5, 6, 7, 8, or

9. Then we must verify that the total number of codes is the same as the number predicted by the multiplication principle.

2. Solution Steps

First, let's list all the possible codes. We will form the codes by pairing each letter with each number.

Codes starting with X:

X5, X6, X7, X8, X9

Codes starting with Y:

Y5, Y6, Y7, Y8, Y9

Codes starting with Z:

Z5, Z6, Z7, Z8, Z9

Now, let's count the total number of codes. There are 5 codes starting with X, 5 codes starting with Y, and 5 codes starting with Z. So the total number of codes is $5+5+5 = 15$.

The multiplication principle states that if there are $n$ ways to do one thing and $m$ ways to do another, then there are $n \times m$ ways to do both. In this problem, there are 3 choices for the first digit (X, Y, or Z) and 5 choices for the second digit (5, 6, 7, 8, or 9). Therefore, the multiplication principle predicts that there will be $3 \times 5 = 15$ possible codes.

Since we found 15 codes by listing them out, and the multiplication principle predicts 15 codes, they agree.

3. Final Answer

The list of 2-digit codes is: X5, X6, X7, X8, X9, Y5, Y6, Y7, Y8, Y9, Z5, Z6, Z7, Z8, Z

9. The total number of codes is

1

5. The multiplication principle predicts 15 codes, which agrees with the number of codes listed.