A photographer will arrange 6 people of 6 different heights for photograph by placing them in two rows of three...

1. Translate the problem requirements: We need to arrange 6 people of different heights in two rows of 3 each, where front row heights increase left to right, back row heights increase left to right, and each back row person is taller than the person directly in front of them.
2. Establish the constraint relationships: Determine what the height ordering constraints mean in terms of which people can be placed in which positions.
3. Use systematic position assignment: Work through the arrangement by fixing certain positions and determining how many ways the remaining positions can be filled.
4. Count valid arrangements: Apply the constraints systematically to count all possible valid arrangements.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to do in everyday language. We have 6 people, all different heights, and we need to arrange them like this:

Back row: Person D | Person E | Person F
Front row: Person A | Person B | Person C

The rules are:
• In the front row: A must be shorter than B, and B must be shorter than C (heights increase left to right)
• In the back row: D must be shorter than E, and E must be shorter than F (heights increase left to right)
• Each person in back must be taller than the person directly in front: \(\mathrm{D} > \mathrm{A}\), \(\mathrm{E} > \mathrm{B}\), \(\mathrm{F} > \mathrm{C}\)

Let's call our 6 people by their heights: 1 (shortest), 2, 3, 4, 5, 6 (tallest).

Process Skill: TRANSLATE - Converting the spatial arrangement and ordering constraints into clear mathematical relationships

2. Establish the constraint relationships

Now let's think about what these constraints mean. We have the overall ordering from our constraints:

\(\mathrm{A} < \mathrm{B} < \mathrm{C}\) (front row increases)
\(\mathrm{D} < \mathrm{E} < \mathrm{F}\) (back row increases)
\(\mathrm{A} < \mathrm{D}\), \(\mathrm{B} < \mathrm{E}\), \(\mathrm{C} < \mathrm{F}\) (back row taller than front row)

Combining these relationships, we can see that the overall height ordering must be:
\(\mathrm{A} < \mathrm{D} < \mathrm{E} < \mathrm{F}\) and \(\mathrm{A} < \mathrm{B} < \mathrm{C} < \mathrm{F}\)

This means person 1 (shortest) must be in position A (front left), and person 6 (tallest) must be in position F (back right).

Process Skill: INFER - Drawing the non-obvious conclusion about fixed positions from the constraint relationships

3. Use systematic position assignment

Since we know person 1 goes in position A and person 6 goes in position F, we need to figure out how to arrange persons 2, 3, 4, and 5 in positions B, C, D, and E.

The remaining constraints are:
• \(1 < \mathrm{B} < \mathrm{C} < 6\) (so B and C must be chosen from {2, 3, 4, 5})
• \(1 < \mathrm{D} < \mathrm{E} < 6\) (so D and E must be chosen from {2, 3, 4, 5})
• \(\mathrm{B} < \mathrm{E}\) and \(\mathrm{C} < \mathrm{F}\) (which is automatically satisfied since F = 6)

So we need to choose 2 people from {2, 3, 4, 5} for positions B and C (with \(\mathrm{B} < \mathrm{C}\)), and the remaining 2 people go to positions D and E (with \(\mathrm{D} < \mathrm{E}\)), such that \(\mathrm{B} < \mathrm{E}\).

4. Count valid arrangements

Let's systematically go through all possibilities for positions B and C, then see what works for D and E:

Case 1: B = 2, C = 3
Remaining people: {4, 5} for positions D, E
We need \(\mathrm{D} < \mathrm{E}\), so D = 4, E = 5
Check: \(\mathrm{B} < \mathrm{E}\)? \(2 < 5\) ✓
Arrangement: A=1, B=2, C=3, D=4, E=5, F=6

Case 2: B = 2, C = 4
Remaining people: {3, 5} for positions D, E
We need \(\mathrm{D} < \mathrm{E}\), so D = 3, E = 5
Check: \(\mathrm{B} < \mathrm{E}\)? \(2 < 5\) ✓
Arrangement: A=1, B=2, C=4, D=3, E=5, F=6

Case 3: B = 2, C = 5
Remaining people: {3, 4} for positions D, E
We need \(\mathrm{D} < \mathrm{E}\), so D = 3, E = 4
Check: \(\mathrm{B} < \mathrm{E}\)? \(2 < 4\) ✓
Arrangement: A=1, B=2, C=5, D=3, E=4, F=6

Case 4: B = 3, C = 4
Remaining people: {2, 5} for positions D, E
We need \(\mathrm{D} < \mathrm{E}\), so D = 2, E = 5
Check: \(\mathrm{B} < \mathrm{E}\)? \(3 < 5\) ✓
Arrangement: A=1, B=3, C=4, D=2, E=5, F=6

Case 5: B = 3, C = 5
Remaining people: {2, 4} for positions D, E
We need \(\mathrm{D} < \mathrm{E}\), so D = 2, E = 4
Check: \(\mathrm{B} < \mathrm{E}\)? \(3 < 4\) ✓
Arrangement: A=1, B=3, C=5, D=2, E=4, F=6

Case 6: B = 4, C = 5
Remaining people: {2, 3} for positions D, E
We need \(\mathrm{D} < \mathrm{E}\), so D = 2, E = 3
Check: \(\mathrm{B} < \mathrm{E}\)? \(4 < 3\) ✗
This case doesn't work!

Process Skill: APPLY CONSTRAINTS - Systematically checking each case against all constraints to ensure validity

4. Final Answer

We found exactly 5 valid arrangements of the 6 people satisfying all the given constraints.

The answer is (A) 5.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the constraint relationships
Students often struggle to correctly translate "each person in the second row must be taller than the person standing in front of him or her" into the mathematical relationship \(\mathrm{D} > \mathrm{A}\), \(\mathrm{E} > \mathrm{B}\), \(\mathrm{F} > \mathrm{C}\). They might think this means the back row people just need to be taller than someone in the front row (not necessarily the person directly in front), leading to incorrect constraint setup.

2. Missing the logical deduction about fixed positions
Many students fail to realize that the combination of all constraints forces person 1 (shortest) to be in position A and person 6 (tallest) to be in position F. Instead, they attempt to solve the problem by considering all possible positions for all people, making the problem much more complex and often leading to incorrect counting.

3. Choosing an overly complex counting approach
Students might try to use permutation formulas or advanced combinatorial methods instead of recognizing this as a systematic case-by-case enumeration problem. This leads them down complicated paths with factorial calculations that don't account for the specific constraints properly.

Errors while executing the approach

1. Forgetting to check the B < E constraint
When systematically going through cases, students often remember to check that \(\mathrm{B} < \mathrm{C}\) and \(\mathrm{D} < \mathrm{E}\) within each row, but forget to verify the crucial cross-row constraint that \(\mathrm{B} < \mathrm{E}\). This leads to including invalid arrangements (like Case 6 in the solution) in their final count.

2. Arithmetic errors in constraint checking
Students may correctly set up a case like "B = 4, C = 5, D = 2, E = 3" but then make simple arithmetic errors when checking whether \(4 < 3\), either rushing through the verification or mixing up which numbers they're comparing.

3. Missing cases or double-counting
When enumerating all possible arrangements systematically, students might skip valid cases (like forgetting B = 3, C = 4) or accidentally count the same arrangement twice by approaching it from different angles, leading to an incorrect total count.

Errors while selecting the answer

No likely faltering points