There are 150 students at Seward High School. 66 students play baseball, 45 play basketball, and 42 play soccer. 27...

1. Translate the problem requirements: We need to find how many students play none of the three sports. We know the total students (150), individual sport participation (66 baseball, 45 basketball, 42 soccer), students playing exactly two sports (27), and students playing all three sports (3).
2. Calculate total sport participations accounting for overlaps: Add up all individual sport counts, then systematically subtract the overlaps to avoid double-counting students who play multiple sports.
3. Apply the inclusion-exclusion principle logically: Use the relationship that students playing at least one sport equals individual totals minus overlaps, considering that students playing all three sports are counted in both the "exactly two sports" overlap and individual sport totals.
4. Find students playing no sports: Subtract the number of students playing at least one sport from the total student population to get those playing none.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for and what information we have.

We want to find: How many students play NONE of the three sports?

What we know:

• Total students at the school: 150
• Students who play baseball: 66
• Students who play basketball: 45
• Students who play soccer: 42
• Students who play exactly two sports: 27
• Students who play all three sports: 3

Think of this like organizing a party invitation list where some people are invited to multiple events. We need to count carefully so we don't double-count anyone.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Calculate total sport participations accounting for overlaps

If we simply add up all the individual sport players, we get:
\(66 + 45 + 42 = 153\) participations

But wait! This number (153) is actually larger than our total student body (150). This tells us that some students are being counted multiple times because they play more than one sport.

Think of it this way: If student Sarah plays both baseball and basketball, she gets counted once in the "66 baseball players" and once again in the "45 basketball players." So in our total of 153, Sarah is counted twice instead of once.

We need to subtract out these multiple countings to get the actual number of students who play at least one sport.

3. Apply the inclusion-exclusion principle logically

Now let's think through the overlaps step by step:

• Students playing exactly two sports: 27
• Students playing all three sports: 3

Here's the key insight: When we calculated 153 total participations, students playing exactly two sports were counted twice (once for each sport they play), and students playing all three sports were counted three times (once for each of the three sports).

To find students playing at least one sport:

• Start with total participations: 153
• Subtract the extra counting for students with exactly two sports: \(153 - 27 = 126\)
• Subtract the extra counting for students with all three sports (they were counted 3 times but should only be counted once, so subtract 2): \(126 - (2 \times 3) = 126 - 6 = 120\)

So 120 students play at least one sport.

Process Skill: INFER - Drawing the non-obvious conclusion about how multiple countings work in overlapping sets

4. Find students playing no sports

Now this is straightforward! If 120 students play at least one sport, then the remaining students play none of the sports.

Students playing no sports = Total students - Students playing at least one sport
Students playing no sports = \(150 - 120 = 30\)

Final Answer

30 students play none of the three sports.

This matches answer choice C.

Let's verify this makes sense: 30 students play no sports + 120 students play at least one sport = 150 total students ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "exactly two sports" constraint

Students often confuse "exactly two sports" with "at least two sports" or "at most two sports." The problem states that 27 students play exactly two sports, which means these students play two sports but NOT three. This distinction is crucial for correctly applying the inclusion-exclusion principle.

2. Attempting to use Venn diagram without understanding overlap relationships

Many students jump straight to drawing a three-circle Venn diagram but fail to properly interpret how the given information (exactly two sports: 27, all three sports: 3) translates to the individual intersection regions in the diagram. They may incorrectly assume these numbers directly represent specific Venn diagram regions.

3. Not recognizing this as a "multiple counting" problem

Students may fail to understand that when we add \(66 + 45 + 42 = 153\), we're counting some students multiple times. They might not realize that this total exceeds the number of students (150), which should be a red flag indicating overlaps need to be addressed.

Errors while executing the approach

1. Incorrect application of inclusion-exclusion adjustments

When removing double/triple counting, students often make errors in determining how many times to subtract. For example, they might subtract 3 (students playing all three sports) only once instead of subtracting \(2 \times 3 = 6\), failing to understand that these students were counted 3 times but should only be counted once.

2. Arithmetic errors in the step-by-step calculation

Students may make simple computational mistakes when performing: \(153 - 27 - 6 = 120\), or in the final subtraction \(150 - 120 = 30\). These arithmetic errors can lead to selecting incorrect answer choices.

3. Confusing the direction of the inclusion-exclusion logic

Some students might add overlaps instead of subtracting them, or subtract when they should add, fundamentally misunderstanding how to correct for multiple counting in overlapping sets.

Errors while selecting the answer

1. Selecting the number who play at least one sport instead of none

After correctly calculating that 120 students play at least one sport, students might accidentally select this value or a close answer choice instead of recognizing they need to find \(150 - 120 = 30\) students who play none of the sports.

2. Choosing the "exactly two sports" value by mistake

Students might see that 27 students play exactly two sports and incorrectly select answer choice B (27), confusing this intermediate value with the final answer about students who play no sports.