A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance...

1. Translate the problem requirements: Convert the given information about people who "cannot" do activities into people who "can" do activities, and identify what we need to find - people who can do none of these activities.
2. Apply the inclusion-exclusion framework: Use the relationship that people doing at least one activity equals the sum of individual activities minus overlaps, working with the constraint that people doing at least two activities is 37.
3. Solve for the overlap between exactly two activities: Use the given information about people doing all three activities (6) and people doing at least two activities (37) to find people doing exactly two activities.
4. Calculate people doing at least one activity: Apply the inclusion-exclusion principle with the known overlaps to find the total number of people who can do at least one of swimming, dancing, or driving.
5. Determine people doing none: Subtract the number of people doing at least one activity from the total population to find people who cannot do any of these activities.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and converting the "cannot do" information into "can do" information.

Given information:

• Total residents: 144
• Cannot swim: 89, so can swim: \(144 - 89 = 55\)
• Cannot dance: 100, so can dance: \(144 - 100 = 44\)
• Cannot drive: 91, so can drive: \(144 - 91 = 53\)
• Can do at least two activities: 37
• Can do all three activities: 6

What we need to find: How many people cannot do any of these three activities (swimming, dancing, driving).

Process Skill: TRANSLATE - Converting "cannot" statements to "can" statements is crucial for applying set theory principles correctly.

2. Apply the inclusion-exclusion framework

Think of this like organizing people into groups. Some people can do one activity, some can do two, and some can do all three. The inclusion-exclusion principle helps us avoid double-counting people who appear in multiple groups.

The basic relationship is:

People doing at least one activity = (Sum of individual activities) - (People doing exactly two) - \(2 \times\)(People doing all three)

We know:

• Sum of individual activities = \(55 + 44 + 53 = 152\)
• People doing all three = 6
• People doing at least two = 37

3. Solve for the overlap between exactly two activities

Here's the key insight: "People doing at least two activities" includes both people doing exactly two activities AND people doing all three activities.

So: People doing exactly two = People doing at least two - People doing all three

People doing exactly two = \(37 - 6 = 31\)

Process Skill: INFER - Recognizing that "at least two" includes "exactly two" plus "all three" is a critical logical step.

4. Calculate people doing at least one activity

Now we can apply the inclusion-exclusion formula:

People doing at least one = \(152 - 31 - 2 \times 6\)

People doing at least one = \(152 - 31 - 12\)

People doing at least one = 109

Let's verify this makes sense: We started with 152 total "activity instances" but subtracted 31 (for people counted twice) and 12 (for people counted three times, so we subtract the extra 2 counts).

5. Determine people doing none

This is straightforward subtraction:

People doing none = Total residents - People doing at least one

People doing none = \(144 - 109 = 35\)

Final Answer

35 people cannot do any of these activities.

This matches answer choice D) 35.

Quick verification: 109 people can do at least one activity + 35 people can do none = 144 total residents ✓

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "at least two" as "exactly two"

Students often confuse the phrase "people who could do at least two of these things" and treat it as "people who could do exactly two things." This leads them to incorrectly use 37 as the number doing exactly two activities, rather than recognizing that 37 includes both people doing exactly two AND people doing all three activities.

Faltering Point 2: Working directly with "cannot do" numbers instead of converting

Many students attempt to work directly with the given "cannot" numbers (89, 100, 91) in set theory formulas. However, inclusion-exclusion principle requires working with positive sets (people who CAN do activities). Failing to convert these to "can do" numbers (55, 44, 53) leads to incorrect setup of the problem.

Faltering Point 3: Incorrect application of inclusion-exclusion formula

Students may use the wrong version of inclusion-exclusion or apply it incorrectly. Some might use |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C| without properly identifying what each intersection represents, leading to confusion about whether to use individual pairwise intersections or total "exactly two" count.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in subtraction and multiplication

Students may make simple calculation errors such as: \(144 - 89 = 54\) instead of 55, or \(37 - 6 = 32\) instead of 31, or \(2 \times 6 = 10\) instead of 12. These small errors cascade through the solution and lead to incorrect final answers.

Faltering Point 2: Incorrectly calculating "exactly two" overlap

Even when students understand that "at least two" includes "all three," they might subtract incorrectly or use the wrong formula. For example, they might calculate \(37 + 6\) instead of \(37 - 6\), thinking they need to add the overlaps rather than isolate the "exactly two" portion.

Errors while selecting the answer

Faltering Point 1: Selecting intermediate calculation as final answer

Students might select 109 (people who can do at least one activity) thinking this is what the question asks for, rather than completing the final step to find \(144 - 109 = 35\) (people who cannot do any activity). They lose track of what the question is actually asking for in the final step.