ExamShortcut
🔍Log in

Venn Diagrams

🔒 Log in to track
medium importance~1-2 Q in Tier 111 formulas⚡ 8 shortcuts4 subtopics
OverviewLearn (4)Formulas (11)Examples (4)Practice (55)
All subtopics·Subtopic 3 of 4

Three-circle counting

🔒 Log in to track
⏱ 5 min read🧩 6 question types🎯 13 practice Q
The idea in one minute

Three circles make eight parts: seven inside and one outside. The union formula adds the singles, subtracts the pairs and adds the triple back. Then 'exactly one', 'exactly two' and 'none' are simple sums.

01

The seven parts

Three groups A, B and C make seven parts inside the circles.

PartMeaning
Only A, only B, only CIn one group only
A and B only, B and C only, A and C onlyIn exactly two groups
All threeIn A, B and C

The outside is the eighth part. It holds people in none of the groups.

02

Why the union formula works

Add the three groups first. People in two groups are counted twice. People in all three are counted three times.

Subtract the three pairs. Now the all-three people are counted 3 − 3 = 0 times. Add the all-three part back once.

n(A∪B∪C)=ΣA−Σpairs+triplen(A \cup B \cup C) = \Sigma A - \Sigma \text{pairs} + \text{triple}

Rule: Union = (A + B + C) − (AB + BC + AC) + ABC. None = total − union.

03

Worked example

There are 60 members. A has 30, B has 25 and C has 20. The pairs are 10, 8 and 7. All three is 4.

Union = 75 − 25 + 4 = 54. None = 60 − 54 = 6.

Tip: Write the seven numbers in a list before you start. This stops missed terms.

04

Filling the parts from the outside in

The pair numbers include the centre part. So remove the triple from each pair before you use it.

  • Only A and B = AB − ABC.
  • Only A = A − (AB − ABC) − (AC − ABC) − ABC, which equals A − AB − AC + ABC.

Example: A = 45, AB = 15, AC = 12, ABC = 6. Only A = 45 − 15 − 12 + 6 = 24.

05

Exactly one, two or three

Each pair figure contains the triple once. So the three pair figures together contain it three times.

  • Exactly two = AB + BC + AC − 3 × ABC.
  • At least two = AB + BC + AC − 2 × ABC.
  • Exactly one = union − exactly two − ABC.

Watch: "Exactly two" is not the sum of the pairs. That sum has the triple counted three times.

Example: pairs 14, 12 and 11, and triple 5. Exactly two = 37 − 15 = 22.

06

Working backwards

Sometimes the union is given and a group total is missing. Rearrange the formula.

ΣA=union+Σpairs−triple\Sigma A = \text{union} + \Sigma \text{pairs} - \text{triple}

Example: union 120, pairs add to 50, triple 20. Then A + B + C = 120 + 50 − 20 = 150.

Note: After you find every part, check that no part is negative. A negative part means a number was misread.

07

Full check on the club example

Use the 60-member club again: A = 30, B = 25, C = 20, pairs 10, 8, 7, triple 4.

  • Only A = 30 − 10 − 7 + 4 = 17. Only B = 25 − 10 − 8 + 4 = 11. Only C = 20 − 8 − 7 + 4 = 9.
  • Exactly one = 17 + 11 + 9 = 37.
  • Exactly two = 25 − 3 × 4 = 13.
  • All three = 4, and none = 6.

Add them: 37 + 13 + 4 + 6 = 60. The parts match the total, so every step was right.

08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Union and none

How to spot it:

Three group totals, three pair overlaps and the triple. It asks for at least one or for none.

n(A∪B∪C)=ΣA−Σpairs+triplen(A ∪ B ∪ C) = ΣA − Σpairs + triple
Method

  1. Add the three groups.

  2. Subtract the three pairs.

  3. Add the triple.

  4. For none, subtract the union from the total.

Why it works:

The triple is added 3 times and taken out 3 times, so add it back once.

Try this

There are 60 members. A has 30, B has 25, C has 20. The pairs are 10, 8 and 7, and all three is 4. How many are in none?

Show solution

  1. Union = 75 − 25 + 4 = 54.

  2. None = 60 − 54 = 6.

Answer

6

Type 2common2 practice Q

Exactly two groups

How to spot it:

The words 'exactly two' or 'only two' with three pair figures and the triple.

Exactlytwo=Σpairs−3×tripleExactly two = Σpairs − 3 × triple
Method

  1. Add the three pair figures.

  2. Subtract 3 times the triple.

  3. Check with the parts if you have time.

Why it works:

Each pair figure includes the triple once, so the sum has it three times.

Try this

In a survey of three groups, the pair overlaps are 14, 12 and 11, and 5 are in all three. How many are in exactly two groups?

Show solution

  1. Sum of pairs = 14 + 12 + 11 = 37.

  2. Remove 3 × 5 = 15.

  3. 37 − 15 = 22.

Answer

22

Type 3common2 practice Q

Find one region

How to spot it:

It asks for 'only A' with A, the two pairs that include A, and the triple.

OnlyA=A−AB−AC+ABCOnly A = A − AB − AC + ABC
Method

  1. Subtract AB from A.

  2. Subtract AC from A.

  3. Add ABC back.

Why it works:

AB and AC both include the triple, so it was subtracted twice.

Try this

A has 45 members. A and B share 15, A and C share 12, and 6 are in all three. How many are only in A?

Show solution

  1. 45 − 15 = 30.

  2. 30 − 12 = 18.

  3. Add back 6: 18 + 6 = 24.

Answer

24

Type 4occasional2 practice Q

Find the total of the groups

How to spot it:

The union, the pair figures and the triple are given. It asks for the sum of the three groups.

ΣA=union+Σpairs−tripleΣA = union + Σpairs − triple
Method

  1. Write the union formula.

  2. Move ΣA to one side.

  3. Put in the numbers.

Why it works:

The union formula is just rearranged.

Try this

The union of three groups is 120. The pairs add up to 50 and the triple is 20. What is A + B + C?

Show solution

  1. ΣA = union + pairs − triple.

  2. = 120 + 50 − 20.

  3. = 150.

Answer

150

Type 5common

Exactly one group

How to spot it:

It asks for members in only one of the three groups.

Exactlyone=union−exactlytwo−tripleExactly one = union − exactly two − triple
Method

  1. Find the union.

  2. Find exactly two.

  3. Subtract exactly two and the triple from the union.

Why it works:

The union is split into exactly one, exactly two and all three.

Try this

For groups with A = 30, B = 25, C = 20, pairs 10, 8, 7 and triple 4, how many are in exactly one group?

Show solution

  1. Union = 75 − 25 + 4 = 54.

  2. Exactly two = 25 − 12 = 13.

  3. Exactly one = 54 − 13 − 4 = 37.

Answer

37

Type 6occasional

At least two groups

How to spot it:

The words 'two or more groups' or 'at least two'.

Atleasttwo=Σpairs−2×tripleAt least two = Σpairs − 2 × triple
Method

  1. Add the three pair figures.

  2. Subtract 2 times the triple.

  3. This equals exactly two plus the triple.

Why it works:

Exactly two is the pairs minus 3 triples. At least two adds the triple back once.

Try this

The pair overlaps are 14, 12 and 11, and 5 are in all three. How many are in at least two groups?

Show solution

  1. Sum of pairs = 37.

  2. Remove 2 × 5 = 10.

  3. 37 − 10 = 27.

Answer

27

09

Formula sheet

Three-set union
∣A∪B∪C∣=Σ∣A∣−Σ∣A∩B∣+∣A∩B∩C∣|A \cup B \cup C| = \Sigma|A| - \Sigma|A \cap B| + |A \cap B \cap C|

Add singles, subtract pairs, add the triple back.

Exactly two
Σ∣A∩B∣−3∣A∩B∩C∣\Sigma|A \cap B| - 3|A \cap B \cap C|

Sum of the pair figures minus three times the triple.

At least two
Σ∣A∩B∣−2∣A∩B∩C∣\Sigma|A \cap B| - 2|A \cap B \cap C|

Exactly two plus the triple.

Exactly one
∣A∪B∪C∣−exactly two−all three|A \cup B \cup C| - \text{exactly two} - \text{all three}

Remove the multi-group members from the union.

10

Shortcuts that save time

⚡ Strip the triple

For 'exactly two', take the pair numbers and remove the triple three times. It works because each pair holds the triple once.

Example

AB = 8, BC = 7, AC = 6 and all three = 3. How many are in exactly two groups?

Show solution

  1. Sum of pairs = 8 + 7 + 6 = 21.

  2. Remove 3 × 3 = 9.

  3. 21 − 9 = 12.

Answer

12

⚡ Fill the centre first

Start with the all-three part. Then fill the pairs, then the singles. Each step needs only the numbers you already have.

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using the pair numbers as 'exactly two'.

Each pair includes the triple. Exactly two = sum of pairs − 3 × triple.

Mistake 02

Forgetting to add the triple back in the union formula.

Union = singles − pairs + triple. The triple must be added once.

Mistake 03

Finding 'only A' as A − AB − AC without the triple.

The triple is subtracted twice there. Add it back: A − AB − AC + ABC.

Mistake 04

Taking the union as the final answer for 'none'.

None = total − union.

Mistake 05

Not checking for a negative part.

Every part must be 0 or more. A negative part means a slip.

12

Quick revision

Read this the night before the exam.

  • Union = A + B + C − AB − BC − AC + ABC.

  • None = total − union.

  • Only A = A − AB − AC + ABC.

  • Exactly two = AB + BC + AC − 3 × ABC.

  • At least two = AB + BC + AC − 2 × ABC.

  • Exactly one = union − exactly two − ABC.

  • A + B + C = union + pairs − triple.

  • Every part must be 0 or more.

13

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 11 min · wrong answers go to your mistake notebook automatically.

Continue learning