Inclusion-Exclusion with Three Sets: Finding Exact Overlaps

Number Puzzles 9th-10th Grade
PROBLEM
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?

What This Problem Teaches

1
Converting percentages to absolute counts in multi-set problems
2
Applying the inclusion-exclusion principle with three overlapping sets
3
Distinguishing between "at least two" and "exactly two" in overlap calculations
4
Setting up and solving equations involving set relationships and constraints
5
Interpreting Venn diagram regions algebraically without drawing the full diagram

Visualizing the Problem

Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30...

This Venn diagram shows three overlapping sets. We know the total size of each circle and the center intersection, but we need to find how many houses fall in exactly the two-way overlaps (excluding the center).

Solution: Method 1 — Inclusion-Exclusion with Variables

Step 1 — Convert percentages to house counts

First, let's convert the given percentages into actual numbers of houses:

Air-conditioning: 60% of 150 = 0.6 × 150 = 90 houses
Sunporch: 50% of 150 = 0.5 × 150 = 75 houses
Swimming pool: 30% of 150 = 0.3 × 150 = 45 houses

Step 2 — Set up the inclusion-exclusion formula

Let's define our variables. We know that 5 houses have none of the amenities, so 145 houses have at least one amenity. The inclusion-exclusion principle states:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

Substituting our known values:

145 = 90 + 75 + 45 - |A ∩ B| - |A ∩ C| - |B ∩ C| + 5

Step 3 — Express overlaps in terms of "exactly two"

Here's the key insight: each two-way intersection consists of houses with exactly those two amenities PLUS the houses with all three. If we let x be the number of houses with exactly two amenities, then:

|A ∩ B| = (houses with exactly A and B) + (houses with all three)
|A ∩ C| = (houses with exactly A and C) + (houses with all three)
|B ∩ C| = (houses with exactly B and C) + (houses with all three)

So the sum of all two-way intersections is:

|A ∩ B| + |A ∩ C| + |B ∩ C| = x + 3(5) = x + 15

Step 4 — Solve for exactly two amenities

Substituting back into our inclusion-exclusion equation:

145 = 210 - (x + 15) + 5
145 = 200 - x
x = 200 - 145 = 55
75 houses have exactly two of these amenities.
Wait! Let me double-check this calculation. I made an error above.

Step 4 (Corrected) — Solve for exactly two amenities

Let me recalculate more carefully. From the inclusion-exclusion formula:

145 = 90 + 75 + 45 - (x + 15) + 5
145 = 210 - x - 15 + 5
145 = 200 - x
x = 200 - 145 = 55

But wait—this gives us the sum of all two-way overlaps excluding the center. We need to be more careful about what x represents.

Let me restart with clearer notation. Let e = exactly two amenities (what we want to find). The sum of the two-way intersections is:

|A ∩ B| + |A ∩ C| + |B ∩ C| = e + 3(5) = e + 15

From inclusion-exclusion:

145 = 210 - (e + 15) + 5
145 = 200 - e
e = 55
55 houses have exactly two of these amenities.

Solution: Method 2 — Region-by-Region Analysis

Step 1 — Define variables for each Venn diagram region

Let's break down the Venn diagram into seven distinct regions plus the "none" category:

  • a = houses with only air-conditioning
  • b = houses with only sunporch
  • c = houses with only swimming pool
  • d = houses with exactly A and B (no pool)
  • e = houses with exactly A and C (no sunporch)
  • f = houses with exactly B and C (no A/C)
  • g = houses with all three = 5
  • None = 5

Step 2 — Set up system of equations

We can write four equations based on what we know:

Total houses: a + b + c + d + e + f + 5 + 5 = 150
A/C total: a + d + e + 5 = 90
Sunporch total: b + d + f + 5 = 75
Pool total: c + e + f + 5 = 45

Step 3 — Simplify the system

From the first equation: a + b + c + d + e + f = 140

From the other equations:

a + d + e = 85
b + d + f = 70
c + e + f = 40

Step 4 — Solve for exactly two amenities

Adding the last three equations:

(a + d + e) + (b + d + f) + (c + e + f) = 85 + 70 + 40
a + b + c + 2d + 2e + 2f = 195
(a + b + c + d + e + f) + (d + e + f) = 195
140 + (d + e + f) = 195
d + e + f = 55

The quantity d + e + f represents exactly the houses with exactly two amenities.

55 houses have exactly two of these amenities.

Verification

Let's verify our answer by checking that all regions sum to 150 houses total.

We found that 55 houses have exactly two amenities. Now we can find how many have exactly one:

Houses with at least one amenity: 150 - 5 = 145
Houses with exactly one amenity: 145 - 55 - 5 = 85

Let's verify this makes sense with our original totals:

Total "amenity instances": 90 + 75 + 45 = 210
From exactly one: 85 × 1 = 85
From exactly two: 55 × 2 = 110
From exactly three: 5 × 3 = 15
Total: 85 + 110 + 15 = 210 ✓

Perfect! Our answer checks out.

Where Students Go Wrong

✗ MISTAKE #1: Adding the percentages

Some students think: "60% + 50% + 30% = 140%, so 140% of 150 = 210 houses have amenities." This is wrong because it counts overlapping houses multiple times. The inclusion-exclusion principle exists precisely to correct this over-counting.

✗ MISTAKE #2: Confusing "exactly two" with "at least two"

Students might calculate that 80 houses have at least two amenities and report this as the final answer. But "at least two" includes houses with all three amenities, while "exactly two" excludes them. Always subtract the triple intersection: 80 - 5 = 75.

✗ MISTAKE #3: Forgetting the "none" category

Some students use 150 as the total for houses with at least one amenity, forgetting that 5 houses have none. The correct total for |A ∪ B ∪ C| is 150 - 5 = 145.

✗ MISTAKE #4: Misunderstanding two-way intersections

When using inclusion-exclusion, |A ∩ B| means all houses with both A and B, including those with C as well. It's not just houses with exactly A and B. This is why each two-way intersection includes houses with all three amenities.

The Pattern Behind This

This problem demonstrates the inclusion-exclusion principle for three sets. The general formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

The key insight is understanding the relationship between set intersections and the "exactly k" quantities:

Sum of two-way intersections = (exactly two) + 3 × (exactly three)
Sum of individual sets = (exactly one) + 2 × (exactly two) + 3 × (exactly three)

For any three-set problem where you know the individual set sizes, the triple intersection, and the total with none, you can always solve for the "exactly two" quantity using this approach.

Extension: This method generalizes to any number of sets, though the algebra becomes increasingly complex. The inclusion-exclusion principle alternates between adding and subtracting terms based on the size of the intersection.

How to Spot This Problem Type

Look for these tell-tale signs that signal a three-set inclusion-exclusion problem:

  • "What percent have..." followed by multiple overlapping categories
  • "Exactly two of the three" or similar language about specific overlap counts
  • Given percentages that add up to more than 100%
  • "None of the above" or "all three" as additional constraints
  • Survey data about preferences, characteristics, or memberships
Hidden versions: Sometimes this problem appears disguised as quality control ("defects A, B, or C"), market research ("prefer brands X, Y, or Z"), or academic data ("taking classes in math, science, or English"). The mathematical structure is identical regardless of the context.

Real Applications

Three-set inclusion-exclusion appears frequently in real-world analysis:

  • Market Research: Understanding customer overlap between product lines, subscription services, or demographic segments
  • Medical Diagnosis: Tracking patients with multiple comorbidities or risk factors across different conditions
  • Quality Control: Analyzing defect patterns when products can fail in multiple independent ways
  • Survey Analysis: Processing responses where participants can select multiple options from overlapping categories

What If?

1
Changed Triple Intersection
Everything stays the same (150 houses, 60% A/C, 50% sunporch, 30% pool, 5 have none) except now 8 houses have all three amenities instead of 5. How many houses have exactly two amenities?
Step 1 — Use the same setup with new triple intersection

We still have 90 houses with A/C, 75 with sunporch, 45 with pool, and 5 with none. But now 8 houses have all three amenities.

Step 2 — Apply inclusion-exclusion

Let e = exactly two amenities. The sum of two-way intersections is e + 3(8) = e + 24.

Step 3 — Solve the equation

145 = 90 + 75 + 45 - (e + 24) + 8
145 = 210 - e - 24 + 8
145 = 194 - e
e = 49

Answer

49 houses have exactly two amenities.

2
Reverse Problem
In a development of 200 houses, 60% have A/C (120 houses), 40% have sunporch (80 houses), 25% have pool (50 houses). If 3 houses have all three amenities and 45 houses have exactly two amenities, how many houses have none of these amenities?
Step 1 — Set up the inclusion-exclusion equation

Let N = houses with none. Then houses with at least one = 200 - N.
Sum of two-way intersections = 45 + 3(3) = 45 + 9 = 54

Step 2 — Apply the formula

200 - N = 120 + 80 + 50 - 54 + 3
200 - N = 250 - 54 + 3
200 - N = 199

Step 3 — Solve for N

N = 200 - 199 = 1

Answer

1 house has none of these amenities.

3
Adding a Fourth Set
Same 150 houses, but now we track a fourth amenity: garage (40% = 60 houses). Still 60% A/C, 50% sunporch, 30% pool. Now 2 houses have all four amenities, 8 have none, and 48 have exactly two amenities. How many have exactly three amenities?
Step 1 — Count amenity instances

Total amenity instances: 90 + 75 + 45 + 60 = 270
Houses with at least one amenity: 150 - 8 = 142

Step 2 — Set up the counting equation

Let x₁ = exactly 1, x₃ = exactly 3.
x₁ + 48 + x₃ + 2 = 142 (total houses)
x₁ + 2(48) + 3x₃ + 4(2) = 270 (total instances)

Step 3 — Solve the system

From first equation: x₁ = 92 - x₃
Substituting: (92 - x₃) + 96 + 3x₃ + 8 = 270
196 + 2x₃ = 270
x₃ = 37

Answer

37 houses have exactly three amenities.

4
Optimization Challenge
Given the same totals (150 houses, 90 A/C, 75 sunporch, 45 pool, 5 none), what is the minimum possible number of houses with exactly two amenities? (Hint: Consider how to maximize the triple intersection while keeping individual totals fixed.)
Step 1 — Find the maximum possible triple intersection

The triple intersection is limited by the smallest individual set: pool with 45 houses. So maximum |A ∩ B ∩ C| = 45.

Step 2 — Check if this is achievable

If 45 houses have all three amenities, then:
Only A/C: 90 - 45 = 45
Only sunporch: 75 - 45 = 30
Only pool: 45 - 45 = 0

Step 3 — Calculate exactly two

Total houses with amenities: 45 + 30 + 0 + \text{exactly two} + 45 = 145
Exactly two: 145 - 120 = 25

Step 4 — Verify using inclusion-exclusion

Sum of two-way intersections: 25 + 3(45) = 160
Check: 145 = 90 + 75 + 45 - 160 + 45 = 95 ✗

This doesn't work! Let me recalculate the maximum systematically using inclusion-exclusion.

Step 5 — Systematic approach

From inclusion-exclusion: e = 200 - 145 - 3t = 55 - 3t where t is the triple intersection.
To minimize e, we maximize t. The constraint is t ≤ 45 and e ≥ 0.
When t = 45: e = 55 - 3(45) = -80 (impossible)
We need e ≥ 0, so 55 - 3t ≥ 0, thus t ≤ 18.33
Maximum integer value: t = 18, giving e = 55 - 54 = 1

Answer

The minimum number of houses with exactly two amenities is 1.

Frequently Asked Questions

Start by defining variables for each set size and known overlaps. Then use the inclusion-exclusion formula: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|. In this problem, we know 150 houses total with 5 having none, so 145 have at least one amenity. We can solve for the sum of two-way overlaps and then find exactly two.
"At least two" includes houses with exactly two amenities PLUS houses with all three. "Exactly two" excludes the triple overlap. If 80 houses have at least two amenities and 5 have all three, then 80 - 5 = 75 houses have exactly two amenities.
Adding percentages double-counts (or triple-counts) the overlapping regions. Here, adding 60% + 50% + 30% = 140% suggests more houses than exist. The inclusion-exclusion principle systematically corrects for this over-counting by subtracting the two-way overlaps and adding back the three-way overlap.
NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-05-30

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