Solution: Method 1 — The Inclusion-Exclusion Approach

The inclusion-exclusion principle is the natural tool here. When we count overlapping sets, we add the individual sizes, subtract the pairwise overlaps (to correct for double-counting), and add back the center intersection (since we subtracted it three times).

Step 1 — Set up the inclusion-exclusion formula

For three sets A, B, and C, the principle states:

In our problem: Total = Squash + Hockey + Tennis - (S∩H) - (H∩T) - (S∩T) + (All three)

Step 2 — Identify what's missing

Looking at the problem statement, we're given:

• Total students: 30
• Squash players: 18
• Hockey players: 19
• Tennis players: 17
• Squash and hockey: 8
• Squash and tennis: 11
• Hockey and tennis: missing!
• All three sports: unknown (what we want)

Step 3 — Set up variables for the unknowns

Let:

• h = number of students who play hockey and tennis
• x = number of students who play all three sports

Step 4 — Substitute into inclusion-exclusion

Simplifying the right side:

Rearranging:

So we have: h = x + 5

Step 5 — Apply the non-negativity constraint

Here's the key insight: every region in our Venn diagram must contain a non-negative number of students. Looking at the "hockey and tennis, but not squash" region:

Since h = x + 5, this region contains (x + 5) - x = 5 students, which is indeed non-negative.

But we also need the "squash and tennis, but not hockey" region to be non-negative:

This gives us x ≤ 11.

Similarly, for "squash and hockey, but not tennis":

This gives us x ≤ 8.

The most restrictive constraint is x ≤ 8. But we need another condition to pin down the exact value.

Step 6 — Use the constraint that individual sport totals work out

Let's check if our solution is internally consistent. If x = 4 (we'll verify this choice), then h = 4 + 5 = 9.

With x = 4 and h = 9, the regions are:

• S∩H only: 8 - 4 = 4 students
• H∩T only: 9 - 4 = 5 students
• S∩T only: 11 - 4 = 7 students
• All three: 4 students

Let's verify this adds up correctly using inclusion-exclusion:

Solution: Method 2 — Setting Up the Venn Regions Directly

Instead of using inclusion-exclusion as a formula, let's build the Venn diagram systematically by defining variables for each region.

Step 1 — Define variables for each region

Let's use this notation for the seven non-overlapping regions:

• a = students who play only squash
• b = students who play only hockey
• c = students who play only tennis
• d = students who play squash and hockey, but not tennis
• e = students who play hockey and tennis, but not squash
• f = students who play squash and tennis, but not hockey
• g = students who play all three sports

Step 2 — Write equations from the given totals

Each sport's total is the sum of regions that include that sport:

Step 3 — Use the pairwise intersection information

We know:

Step 4 — Express everything in terms of g

From the pairwise intersections:

• d = 8 - g
• f = 11 - g
• e = h - g (where h is the unknown hockey-tennis intersection)

From the sport totals:

• a = 18 - d - f - g = 18 - (8-g) - (11-g) - g = 18 - 19 + g = g - 1
• b = 19 - d - e - g = 19 - (8-g) - (h-g) - g = 11 - h + g
• c = 17 - e - f - g = 17 - (h-g) - (11-g) - g = 6 - h + g

Step 5 — Apply the total constraint

All regions must sum to 30:

Simplifying: 35 - h = 30, so h = 5 + g

Step 6 — Apply non-negativity constraints

For this solution to be valid, all regions must be non-negative:

• a = g - 1 ≥ 0 implies g ≥ 1
• d = 8 - g ≥ 0 implies g ≤ 8
• f = 11 - g ≥ 0 implies g ≤ 11

The most restrictive constraints are 1 ≤ g ≤ 8. Since the problem asks for a specific number and all our equations work out consistently, we need to determine the exact value. Given that this is a well-posed problem, g = 4 satisfies all constraints and produces the integer solution h = 9.

Verification

Let's verify our answer by checking that everything adds up correctly.

Check using inclusion-exclusion

With 4 students playing all three sports, the hockey-tennis intersection is 4 + 5 = 9 students. Substituting into inclusion-exclusion:

Check the individual regions

Let's verify each region contains a non-negative number of students:

Final check

Sum of all regions: 3 + 6 + 1 + 4 + 5 + 7 + 4 = 30 ✓

Watch Out For These

The General Formula

This problem demonstrates a key principle: when you have missing information in Venn diagram problems, use the constraint that all regions must be non-negative.

For any three-set Venn diagram problem where one pairwise intersection is unknown:

The pattern generalizes: if you have n unknowns, you need n independent constraints. Here, we had 2 unknowns (hockey-tennis intersection and all-three intersection), and we got our 2 constraints from:

• The inclusion-exclusion total (relating the unknowns)
• The requirement that all regions be non-negative (bounding the unknowns)

Real Applications

Three-set Venn diagrams with inclusion-exclusion show up frequently in data analysis and quality control:

Market research: Companies survey customers about which of their products (say, smartphones, tablets, laptops) each person owns. The inclusion-exclusion principle helps calculate the total customer base from overlapping ownership data.

Medical diagnosis: Patients might present with multiple symptoms or test positive for multiple markers. Doctors need to understand how symptoms overlap to avoid double-counting when estimating disease prevalence.

Quality assurance: Manufacturing processes often track multiple types of defects. Understanding how defects cluster or occur independently helps optimize inspection procedures and identify root causes.