A palindrome is a number that reads the same forward and backward. For example. 2442 and 111 are palindromes. If...

1. Translate the problem requirements: A 5-digit palindrome reads the same forward and backward (like abcba), and we need to count how many can be formed using digits 1, 2, and 3 with repetition allowed.
2. Identify the palindrome structure: Determine what positions in a 5-digit palindrome are independent choices versus dependent on other positions.
3. Apply the multiplication principle: Count the number of choices for each independent position and multiply to get the total combinations.
4. Verify with the answer choices: Confirm our calculated result matches one of the given options.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. A palindrome reads the same forward and backward. For example, if we have the number 12321, reading it from left to right gives us 1-2-3-2-1, and reading it from right to left also gives us 1-2-3-2-1.

We need to find how many 5-digit palindromes we can create using only the digits 1, 2, and 3. The key point is that we can use these digits with repetition - meaning we can use the same digit multiple times in our palindrome.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Identify the palindrome structure

Let's think about what a 5-digit palindrome looks like. If we call the positions A-B-C-D-E, then for it to read the same forward and backward:

• Position A must equal position E
• Position B must equal position D
• Position C can be anything (it's in the middle)

So our palindrome structure is: A-B-C-B-A

This means we only need to make independent choices for positions A, B, and C. Once we choose these, positions D and E are automatically determined.

For example, if we choose A=1, B=2, C=3, our palindrome becomes 1-2-3-2-1.

Process Skill: VISUALIZE - Seeing the palindrome pattern helps us understand which positions are independent

3. Apply the multiplication principle

Now let's count our choices step by step:

For position A (first digit): We can choose from 1, 2, or 3
Number of choices = 3

For position B (second digit): We can choose from 1, 2, or 3
Number of choices = 3

For position C (middle digit): We can choose from 1, 2, or 3
Number of choices = 3

For positions D and E: These are automatically determined by our choices for B and A respectively, so no additional choices needed.

Using the multiplication principle (when we make independent choices, we multiply the number of options):
Total palindromes = \(3 \times 3 \times 3 = 27\)

4. Verify with the answer choices

Our calculated result is 27 palindromes.

Looking at the answer choices:

1. 12
2. 15
3. 18
4. 24
5. 27

Our answer of 27 matches choice E.

Final Answer

The number of 5-digit palindromes that can be formed using the digits 1, 2, and 3 is 27.

The answer is E) 27.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding palindrome structure for 5-digit numbers
Students often fail to recognize that in a 5-digit palindrome (A-B-C-D-E), only the first three positions (A, B, C) require independent choices. They might think all 5 positions need separate decisions, leading to incorrect counting approaches. The key insight is that positions D and E are automatically determined by positions B and A respectively.

2. Overlooking the "one or more digits" constraint
The problem states palindromes are formed using "one or more of the digits 1, 2, 3." Students might misinterpret this as requiring all three digits to be used in each palindrome, when it actually means any combination of these digits (including repetition) is allowed. This misunderstanding would lead to unnecessary restrictions in their counting approach.

3. Confusing permutation vs. combination thinking
Some students might approach this as a permutation problem, trying to arrange digits in specific orders, rather than recognizing it as a straightforward counting problem where each independent position has a fixed number of choices.

Errors while executing the approach

1. Arithmetic errors in multiplication
Even with the correct approach of \(3 \times 3 \times 3\), students may make simple calculation errors, perhaps computing \(3^2 = 9\) and then adding 3 to get 12, or making other multiplication mistakes that lead to incorrect intermediate results.

2. Incorrect application of the multiplication principle
Students might incorrectly add instead of multiply the choices (\(3 + 3 + 3 = 9\)) or apply the multiplication principle incorrectly by including the dependent positions D and E, calculating \(3^5 = 243\) instead of \(3^3 = 27\).

Errors while selecting the answer

No likely faltering points - the calculation directly produces 27, which clearly matches answer choice E. The answer selection is straightforward once the correct calculation is performed.