Each of the nine digits 0, 1, 1, 4, 5, 6, 8, 8, text{ and } 9 is used once...

1. Translate the problem requirements: We have 9 digits (0, 1, 1, 4, 5, 6, 8, 8, 9) that must be used exactly once to create three 3-digit numbers. We need to find the arrangement that gives the maximum possible sum.
2. Apply place value optimization strategy: To maximize the sum, we need to place the largest available digits in the hundreds places first, then the tens places, then the units places, since hundreds contribute 100 times more than units to the total sum.
3. Handle the constraint of valid 3-digit numbers: Ensure that none of our three numbers starts with 0, as this would create a 2-digit number instead of a 3-digit number.
4. Systematically assign digits by place value priority: Work through hundreds, tens, and units positions in order, always choosing the largest available digits while respecting the 3-digit constraint.
5. Calculate and verify the maximum sum: Add up the three optimized numbers to get our final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we have and what we need to do:

What we have: Nine digits: 0, 1, 1, 4, 5, 6, 8, 8, 9

What we need to do: Use each digit exactly once to create three 3-digit numbers, then find the arrangement that gives us the highest possible sum.

Think of it like having 9 cards with these digits written on them, and we need to arrange them into three piles of 3 cards each to make three numbers. We want the biggest total when we add those three numbers together.

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

2. Apply place value optimization strategy

Here's the key insight: In any number, the hundreds digit contributes 100 times more to the value than the units digit.

For example, compare 851 and 158:

• \(851 = 800 + 50 + 1\)
• \(158 = 100 + 50 + 8\)

Even though they use the same digits, 851 is much larger because 8 is in the hundreds place.

Strategy: To maximize our sum, we should:

• Put the three largest digits in the hundreds places
• Put the next three largest in the tens places
• Put the remaining three in the units places

Our digits in descending order: 9, 8, 8, 6, 5, 4, 1, 1, 0

3. Handle the constraint of valid 3-digit numbers

We have one important rule: none of our numbers can start with 0, because that would make it a 2-digit number, not a 3-digit number.

Since 0 is our smallest digit, this actually works in our favor - we definitely don't want 0 in a hundreds place anyway when we're trying to maximize the sum.

Process Skill: APPLY CONSTRAINTS - Ensuring our solution meets all problem requirements

4. Systematically assign digits by place value priority

Let's assign our digits step by step:

Step 1 - Hundreds places (most important):
Take the three largest digits: 9, 8, 8
So our three numbers start as: 9**, 8**, 8__

Step 2 - Tens places (second most important):
From remaining digits (6, 5, 4, 1, 1, 0), take the three largest: 6, 5, 4
So our numbers become: 96_, 85_, 84_

Step 3 - Units places (least important):
Remaining digits: 1, 1, 0
Assign these: 961, 851, 840

Our three numbers are: 961, 851, 840

5. Calculate and verify the maximum sum

Let's add up our three optimized numbers:

\(961 + 851 + 840\)

Adding step by step:

• \(961 + 851 = 1,812\)
• \(1,812 + 840 = 2,652\)

Let's verify this uses all digits exactly once:

• 961 uses: 9, 6, 1
• 851 uses: 8, 5, 1
• 840 uses: 8, 4, 0

Total digits used: 9, 8, 8, 6, 5, 4, 1, 1, 0 ✓

This matches our original set of digits exactly.

Final Answer

The greatest possible sum is 2,652.

Looking at our answer choices, this corresponds to choice C. 2,652.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the optimization strategy
Students often fail to recognize that place value is the key to maximization. Instead of systematically placing the largest digits in hundreds places, they might try random arrangements or focus on creating one very large number (like 986) while neglecting the others. The critical insight is that maximizing the sum requires optimizing ALL three numbers simultaneously by prioritizing higher place values.

2. Overlooking the constraint about valid 3-digit numbers
While the solution correctly notes that numbers cannot start with 0, students might panic about this constraint or fail to recognize that it actually helps their strategy. Since 0 is the smallest digit, putting it in a hundreds place would hurt the sum anyway. Students may waste time worrying about this constraint instead of seeing it as beneficial.

3. Confusion about digit usage and repetition
The digit set contains repeated values (two 1's and two 8's), which can confuse students. They might incorrectly think they need to use distinct digits only, or conversely, assume they can use any digit multiple times. The key is understanding that each digit in the given set must be used exactly once, regardless of repetition.

Errors while executing the approach

1. Incorrect sorting and assignment of digits
Even with the right strategy, students often make errors in systematically assigning digits by place value. They might incorrectly sort the digits (forgetting about the repeated 8's or 1's) or assign them to the wrong places. For example, accidentally putting a 6 in the hundreds place while leaving an 8 for the tens place.

2. Arithmetic errors in final calculation
When adding 961 + 851 + 840, students frequently make computational mistakes. Common errors include: incorrectly adding the hundreds column (9 + 8 + 8 = 25, carrying the 2), mistakes in the tens column with carrying, or simple addition errors that lead to sums like 2,562 or 2,762 instead of 2,652.

Errors while selecting the answer

No likely faltering points
Once students have correctly calculated 2,652, this value directly corresponds to answer choice C. There are no additional transformations or interpretations required that would typically cause errors in answer selection.