In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone,...

1. Translate the problem requirements: We need to find the maximum (x) and minimum (y) number of households that could have all three devices (DVD player, cell phone, and MP3 player), then calculate x - y.
2. Determine the maximum overlap scenario: Find the largest possible number of households that could have all three devices by identifying what limits this maximum.
3. Determine the minimum overlap scenario: Find the smallest possible number of households that must have all three devices by using the principle that overlaps are forced when the total exceeds the population.
4. Calculate the difference: Subtract the minimum from the maximum to get our final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

We have a village with exactly 100 households. Think of this as 100 boxes, and we're looking at what electronic devices each household owns:

• 75 households have at least one DVD player
• 80 households have at least one cell phone
• 55 households have at least one MP3 player

Our goal is to find two things:

• x = the maximum possible number of households that could have all three devices
• y = the minimum possible number of households that must have all three devices

Then we need to calculate x - y.

Process Skill: TRANSLATE - Converting the overlap language into concrete scenarios we can analyze

2. Determine the maximum overlap scenario

To find the maximum number of households that could have all three devices, we need to think: what limits how high this number can go?

The answer is simple - we can't have more households with all three devices than the smallest group!

Looking at our numbers:

• 75 have DVD players
• 80 have cell phones
• 55 have MP3 players

The smallest group is the 55 households with MP3 players. Even in the most optimistic scenario where every single household that has an MP3 player also happens to have both a DVD player and a cell phone, we still can't exceed 55 households with all three devices.

Therefore: x = 55

Process Skill: CONSIDER ALL CASES - Identifying the constraint that limits the maximum overlap

3. Determine the minimum overlap scenario

Now for the minimum - this is where it gets interesting! We need to find how many households must have all three devices, no matter how we arrange things.

Let's think about this step by step. We have 100 households total, but we're told that:

• 75 have DVD players (so 25 don't have DVD players)
• 80 have cell phones (so 20 don't have cell phones)
• 55 have MP3 players (so 45 don't have MP3 players)

Now, let's count the maximum number of households that could be missing at least one of these devices:

• At most 25 could be missing a DVD player
• At most 20 could be missing a cell phone
• At most 45 could be missing an MP3 player

In the best case scenario (for minimizing overlap), these would all be different households. So the maximum number of households that could be missing at least one device is: \(25 + 20 + 45 = 90\)

But wait! We only have 100 households total. If at most 90 households could be missing at least one device, then at least \(100 - 90 = 10\) households must have all three devices.

Therefore: y = 10

Process Skill: INFER - Using the constraint of total population to force minimum overlap

4. Calculate the difference

Now we simply subtract to find our answer:

\(\mathrm{x} - \mathrm{y} = 55 - 10 = 45\)

Final Answer: C. 45

This makes sense - depending on how the device ownership overlaps, we could have anywhere from 10 to 55 households with all three devices, giving us a range of 45.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what x and y represent
Students often confuse the maximum/minimum number of households with all three devices with other overlap scenarios, such as households with at least two devices or exactly three devices.

2. Incorrectly applying set theory formulas
Many students immediately jump to using Venn diagram formulas like \(|\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}| = |\mathrm{A}| + |\mathrm{B}| + |\mathrm{C}| - |\mathrm{A} \cap \mathrm{B}| - |\mathrm{A} \cap \mathrm{C}| - |\mathrm{B} \cap \mathrm{C}| + |\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}|\), not realizing this approach is unnecessarily complex for finding just the range of the triple intersection.

3. Not recognizing the constraint-based approach
Students may not realize that the maximum overlap is simply limited by the smallest set, and that the minimum overlap can be found using the principle of inclusion-exclusion in reverse.

Errors while executing the approach

1. Incorrect calculation of households missing each device
When finding the minimum overlap, students may forget to subtract from 100 to find how many households DON'T have each device (e.g., calculating 75 instead of \(100 - 75 = 25\) for households without DVD players).

2. Adding overlapping "missing" households incorrectly
Students may incorrectly assume that the maximum number of households missing at least one device equals exactly 25+20+45=90, without understanding this represents the upper bound when all missing households are different.

3. Arithmetic errors in final calculation
Simple computational mistakes when calculating 100-90=10 for the minimum, or 55-10=45 for the final answer.

Errors while selecting the answer

1. Selecting x or y instead of x-y
Students may correctly calculate x=55 and y=10 but then select answer choice B (55) or mistakenly think 10 corresponds to one of the given options, forgetting the question asks for the difference x-y.

2. Confusing maximum and minimum
Students may correctly find both values but then calculate \(\mathrm{y} - \mathrm{x} = 10 - 55 = -45\), leading to confusion about which answer choice to select since negative values aren't offered.