- Translate the problem requirements: We need three-digit numbers where all digits are 1-9 (no zeros), exactly two digits are the same, and one digit is different from the other two (like 112, 343, 787)
- Identify the structural patterns: Recognize that there are exactly three possible arrangements - the repeated digit can be in positions 1&2, 1&3, or 2&3
- Count choices for each pattern systematically: For each arrangement pattern, determine how many ways to choose the repeated digit and the different digit
- Apply the multiplication principle: Multiply the number of patterns by the choices available for each digit type to get the total count
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we're looking for in simple terms. We need three-digit numbers where:
- All digits are between 1 and 9 (no zeros allowed)
- Exactly two digits are the same
- One digit is different from the other two
For example: 112, 343, 787, 565, 898 are all valid because they each have exactly two matching digits and one different digit.
Invalid examples would be: 111 (all three digits same), 123 (all three digits different), 102 (contains zero).
Process Skill: TRANSLATE - Converting the word problem into clear mathematical requirements
2. Identify the structural patterns
Now let's think about where the repeated digit can appear in our three-digit number. If we call our number ABC (where A is hundreds, B is tens, C is units), then the repeated digit can be in exactly three different patterns:
- Pattern 1: \(\mathrm{A = B ≠ C}\) (like 112, 334, 557)
- Pattern 2: \(\mathrm{A = C ≠ B}\) (like 121, 343, 575)
- Pattern 3: \(\mathrm{B = C ≠ A}\) (like 211, 433, 755)
These are the only three ways to arrange exactly two matching digits in a three-digit number.
Process Skill: VISUALIZE - Seeing the different structural arrangements clearly
3. Count choices for each pattern systematically
Let's count the possibilities for each pattern. Since all three patterns work the same way, we can focus on one pattern first.
For Pattern 1 (\(\mathrm{A = B ≠ C}\)):
- Choose the repeated digit (A = B): We can pick any digit from 1 to 9, so that's 9 choices
- Choose the different digit (C): We can pick any digit from 1 to 9, except it must be different from our repeated digit, so that's 8 choices
- Total for Pattern 1: \(\mathrm{9 × 8 = 72}\) numbers
The same logic applies to Pattern 2 and Pattern 3. Each pattern gives us exactly 72 possible numbers.
Process Skill: CONSIDER ALL CASES - Making sure we count each arrangement pattern separately
4. Apply the multiplication principle
Now we combine our results:
- Pattern 1 (\(\mathrm{A = B ≠ C}\)): 72 numbers
- Pattern 2 (\(\mathrm{A = C ≠ B}\)): 72 numbers
- Pattern 3 (\(\mathrm{B = C ≠ A}\)): 72 numbers
Since these patterns are completely separate (no number can fit into more than one pattern), we add them up:
Total = \(\mathrm{72 + 72 + 72 = 216}\)
Final Answer
The total number of three-digit positive integers with no zeros that have exactly two equal digits is 216.
This matches answer choice E. 216.
Common Faltering Points
Errors while devising the approach
- Misinterpreting "two digits equal to each other": Students may think this means "at least two digits equal" and include cases like 111, 222, etc. where all three digits are the same. The problem specifically asks for exactly two equal digits with the third being different.
- Overlooking the "no zeros" constraint: Students might include numbers with zeros (like 110, 100, 202) in their counting, forgetting that the problem explicitly states "no digits equal to zero."
- Missing structural patterns: Students may only consider one arrangement pattern (like \(\mathrm{A=B≠C}\)) and forget that the repeated digits can appear in different positions, missing the other two patterns (\(\mathrm{A=C≠B}\) and \(\mathrm{B=C≠A}\)).
Errors while executing the approach
- Incorrect counting of digit choices: When counting the different digit, students might forget to exclude the repeated digit from their choices, incorrectly using 9 choices instead of 8 for the different digit.
- Double-counting or missing cases: Students might incorrectly think that some numbers belong to multiple patterns (like counting 121 in both \(\mathrm{A=C≠B}\) and \(\mathrm{A=B≠C}\) patterns) or miss some valid combinations entirely.
Errors while selecting the answer
- Forgetting to add all pattern results: Students might correctly calculate 72 for one pattern but forget to multiply by 3 or add the results from all three patterns, stopping at 72 instead of reaching the final answer of 216.
