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A palindrome is a number that reads the same forward and backward. For example. \(\mathrm{2442}\) and \(\mathrm{111}\) are palindromes. If \(\mathrm{5}\)-digit palindromes are formed using one or more of the digits, \(\mathrm{1, 2, 3}\), how many such palindromes are possible?
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
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:
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
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 = \(3 \times 3 \times 3 = 27\)
Our calculated result is 27 palindromes.
Looking at the answer choices:
Our answer of 27 matches choice E.
The number of 5-digit palindromes that can be formed using the digits 1, 2, and 3 is 27.
The answer is E) 27.
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.
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 \(3^3 = 27\).
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.