Each participant in a certain study was assigned a sequence of 3 different letters from the set {A, B, C,...

1. Translate the problem requirements: We need to understand that we're creating sequences of 3 different letters from 8 available letters, where order matters (permutations), and we need to find how many participants received sequences
2. Calculate total possible sequences: Determine how many different 3-letter sequences can be formed from the 8 available letters
3. Apply the given constraint: Use the fact that 36 sequences were not assigned to find how many were actually assigned
4. Connect sequences to participants: Since each participant gets exactly one unique sequence, the number of assigned sequences equals the number of participants

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to do in plain English. We have 8 letters to choose from: A, B, C, D, E, F, G, H. Each participant gets a sequence of exactly 3 different letters from this set.

The key insight here is understanding what makes sequences different. The problem specifically tells us that A, B, C is different from C, B, A. This means the ORDER matters - we're not just picking 3 letters, we're arranging them in a specific sequence.

Think of it like creating license plates with 3 letters - ABC-123 is completely different from CBA-123 even though they use the same letters.

Also important: each sequence goes to exactly one participant (no sharing), and we're told that 36 possible sequences were never assigned to anyone.

Process Skill: TRANSLATE - Converting the problem's language about "sequences" and "different letters" into our mathematical understanding of ordered arrangements

2. Calculate total possible sequences

Now let's figure out how many different 3-letter sequences we can make from our 8 available letters.

Let's think through this step by step:

• For the first position in our sequence, we can choose any of the 8 letters
• For the second position, we can choose any of the remaining 7 letters (since we need different letters)
• For the third position, we can choose any of the remaining 6 letters

So the total number of possible sequences is: \(8 \times 7 \times 6\)

Let's calculate this: \(8 \times 7 = 56\), then \(56 \times 6 = 336\)

This means there are 336 total possible different sequences we could create.

3. Apply the given constraint

The problem tells us that 36 of the possible sequences were NOT assigned to participants.

Since we calculated that there are 336 total possible sequences, and 36 were not assigned, we can find how many WERE assigned by simple subtraction:

Sequences that were assigned = Total possible sequences - Sequences not assigned
Sequences that were assigned = \(336 - 36 = 300\)

4. Connect sequences to participants

The final step is recognizing what the number of assigned sequences tells us about participants.

Since the problem states that "no sequence was assigned to more than one participant," this means each participant gets exactly one unique sequence, and no sequence is shared.

Therefore: Number of participants = Number of assigned sequences = 300

Final Answer

The number of participants in the study was 300.

Looking at our answer choices:

1. 20
2. 92
3. 300
4. 372
5. 476

Our answer of 300 matches choice C exactly.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the meaning of "sequence"
Students often confuse sequences with combinations and fail to recognize that order matters. They might think that A, B, C is the same as C, B, A, even though the problem explicitly states these are different. This leads them to calculate combinations (\(\mathrm{C}(8,3) = 56\)) instead of permutations (\(\mathrm{P}(8,3) = 336\)), resulting in a completely wrong foundation for the entire solution.

Faltering Point 2: Misinterpreting "3 different letters"
Some students might think they can repeat letters in the sequence, missing the constraint that all 3 letters must be different. This would lead them to calculate \(8^3 = 512\) total possibilities instead of the correct \(8 \times 7 \times 6 = 336\), because they'd allow sequences like A, A, B or C, C, C.

Faltering Point 3: Confusion about the constraint "36 sequences were not assigned"
Students might misinterpret this key information, thinking that 36 sequences were assigned rather than understanding that 36 were NOT assigned. This reversal would lead them to conclude there are 36 participants instead of understanding that they need to subtract 36 from the total to find the assigned sequences.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in calculating permutations
Even with the correct approach, students often make simple calculation mistakes when computing \(8 \times 7 \times 6\). Common errors include: \(8 \times 7 = 54\) (instead of 56), or \(56 \times 6 = 330\) (instead of 336). These small arithmetic mistakes cascade through the rest of the solution.

Faltering Point 2: Incorrect subtraction when applying the constraint
Students might add instead of subtract when dealing with the "36 not assigned" constraint, calculating \(336 + 36 = 372\) instead of \(336 - 36 = 300\). This error occurs because they might think the 36 represents additional sequences or participants rather than sequences that were excluded.

Errors while selecting the answer

Faltering Point 1: Selecting the wrong calculated value
Students who correctly calculate both 336 (total sequences) and 300 (assigned sequences) might accidentally select the total number of possible sequences (336) instead of the number of assigned sequences (300) as their final answer. Since 336 isn't among the choices, they might round to the closest option like 300 or 372, potentially getting lucky or unlucky with their mistake.