Chapter 12 of 12

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RRB NTPC General Intelligence & Reasoning★ Roughly 6% of General Intelligence & Reasoning's 30 questions in CBT 1 (prisma/seed-data/exam-blueprints/rrb-ntpc-2026.json → syllabus.subjects[General Intelligence & Reasoning].topics[Venn Diagrams])

Venn Diagrams

Venn diagram questions are counting problems in disguise. The inclusion-exclusion formula handles overlap correctly every time — the recurring trap is double-counting the overlap or forgetting to subtract it at all.

⏱ 8 min readMedium difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

RRB NTPC PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Apply the two-set inclusion-exclusion formula to find 'either,' 'only,' and 'neither' counts
2Apply the three-set inclusion-exclusion formula, correctly adding back the triple overlap
3Identify the correct diagrammatic representation of a three-category relationship
4Compute 'none' counts reliably by first finding 'at least one' via inclusion-exclusion

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Why this chapter matters in RRB NTPC

Nearly every Venn diagram question is a counting problem involving overlapping groups, and the inclusion-exclusion formula resolves both the two-set and three-set cases cleanly and reliably without needing to draw an actual diagram — the recurring trap is mishandling the overlap term.

Before you start — revise these

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Basic arithmetic

This chapter requires only addition and subtraction, applied carefully according to the inclusion-exclusion formulas.

Venn Diagrams — RRB NTPC Reasoning

This topic carries roughly 6% of General Intelligence & Reasoning's 30 questions. Nearly every question is a counting problem involving overlapping groups — the inclusion-exclusion formula resolves the two-set and three-set cases cleanly, without needing to draw an actual diagram.

1. What RRB NTPC actually asks

Expect two-set overlap counting problems (people who like both, either, or neither of two things), three-set overlap counting problems, and occasionally purely diagrammatic questions (which Venn diagram correctly represents the relationship between three given categories, like "Men, Doctors, Fathers").

2. The two-set inclusion-exclusion formula

Either A or B (or both) = A + B − Both. This formula avoids double-counting the people who belong to both groups.

Only A (not B) = A − Both. Neither A nor B = Total − (Either A or B).

3. The three-set inclusion-exclusion formula

At least one of A, B, C = A + B + C − (A∩B) − (B∩C) − (A∩C) + (A∩B∩C).

Each pairwise overlap is subtracted once (to undo double-counting), and the triple overlap is added back (since it was subtracted three times in the pairwise step, but should only be excluded... included... exactly once net).

Only A (none of B or C) = A − (A∩B) − (A∩C) + (A∩B∩C).

4. Diagrammatic (category relationship) questions

For questions asking "which diagram represents Men, Doctors, Fathers," identify the actual logical relationship: some men are doctors (partial overlap), some men are fathers (partial overlap), some doctors are fathers (partial overlap) — and all three categories can overlap with each other without any being a subset of another.

The correct diagram shows three partially overlapping circles, not one contained inside another. A contained-circle diagram is only correct when one category is genuinely a subset of another, like "Men, Mammals, Doctors," where Men is entirely inside Mammals.

Worked examples

Question 1 of 2

Q1. In a class of 50 students, 30 play cricket, 25 play football, and 15 play both. How many students play neither sport?

Pick an option to check your answer.

Show explanation

Solution. Either cricket or football (or both) = 30 + 25 − 15 = 40. Neither = total − either = 50 − 40 = 10.

Answer: (b).

Question 2 of 2

Q2. In a survey of 100 people: 45 play cricket, 38 play hockey, 35 play football, 15 play cricket and hockey, 12 play hockey and football, 10 play cricket and football, and 5 play all three sports. How many people play none of the three sports?

Pick an option to check your answer.

Show explanation

Solution. At least one sport = C+H+F − (C∩H) − (H∩F) − (C∩F) + (C∩H∩F) = 45+38+35 − 15−12−10 + 5 = 118 − 37 + 5 = 86.

None = total − at least one = 100 − 86 = 14. Answer: (c).

6. Common traps

Forgetting to subtract the overlap when finding "either A or B." Simply adding A + B double-counts everyone in both groups.
Forgetting to add back the triple overlap in the three-set formula. Subtracting all three pairwise overlaps removes the triple-overlap group three times, so it must be added back once to net out correctly.
Confusing "only A" with "A." "Only A" specifically excludes anyone who also belongs to another group — always subtract the relevant overlaps.
Misjudging category relationships in diagrammatic questions. Not every three-category relationship is three separate overlapping circles — some categories are genuinely subsets of others (e.g., "Men" is entirely inside "Mammals").

7. Guessing strategy

For "how many play neither/none" questions, computing "at least one" first via inclusion-exclusion and then subtracting from the total is more reliable than trying to reason about the "neither" region directly.

Summary

Venn Diagrams is roughly 6% of General Intelligence & Reasoning's 30 CBT 1 questions.
Two-set formula: Either A or B = A + B − Both. Neither = Total − Either.
Three-set formula: At least one = A+B+C − (pairwise overlaps) + (triple overlap).
"Only A" always means A minus every overlap A has with other groups.
For diagrammatic questions, verify whether categories are genuinely subsets of each other or merely partially overlapping.
Compute "at least one" first, then subtract from the total for "none" — more reliable than reasoning about the excluded region directly.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Two-set: Either A or B

Either A or B (or both) = A + B − Both

Subtracting the overlap avoids double-counting.

Two-set: Only A / Neither

Only A = A − Both. Neither A nor B = Total − (Either A or B).

Three-set: At least one

At least one of A, B, C = A+B+C − (A∩B) − (B∩C) − (A∩C) + (A∩B∩C)

Each pairwise overlap subtracted once; the triple overlap added back once to net out correctly.

Three-set: Only A

Only A (none of B or C) = A − (A∩B) − (A∩C) + (A∩B∩C)

RRB NTPC test series

Ready to be tested on this?

Venn Diagrams shows up in every RRB NTPC paper — sit a timed one and find out where you actually stand.

~2Qs in CBT 1

Mediumpriority

Full RRB NTPC mocks, real clock
General Intelligence & Reasoning-only sectionals & chapter tests
All-India rank + topic-level analysis

Free forever · solver-verified questions

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Traps RRB NTPC sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Forgetting to subtract the overlap when finding 'either A or B'

✓ Simply adding A + B double-counts everyone in both groups — always subtract 'Both' once.

WATCH OUT

✗ Forgetting to add back the triple overlap in the three-set formula

✓ Subtracting all three pairwise overlaps removes the triple-overlap group three times over — it must be added back once to net out correctly.

WATCH OUT

✗ Confusing 'only A' with simply 'A'

✓ 'Only A' specifically excludes anyone who also belongs to another group — always subtract the relevant overlap(s).

WATCH OUT

✗ Assuming all three-category relationships are three overlapping circles

✓ Some categories are genuinely subsets of others (e.g., 'Men' entirely inside 'Mammals') — verify the actual logical relationship before choosing a diagram.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Venn Diagrams?

8 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

8 questions~6 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Venn Diagrams is roughly 6% of General Intelligence & Reasoning's 30 CBT 1 questions.
•Two-set formula: Either A or B = A + B − Both. Neither = Total − Either.
•Three-set formula: At least one = A+B+C − (pairwise overlaps) + (triple overlap).
•'Only A' always means A minus every overlap A has with other groups.
•Compute 'at least one' first, then subtract from the total for 'none' — more reliable than reasoning about the excluded region directly.
•Verify whether categories are genuinely subsets of each other before choosing a diagrammatic representation.

RRB NTPC question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: Roughly 6% of General Intelligence & Reasoning's 30 questions in CBT 1 (each worth +1/−1/3)

Question style	Marks each	Typical count	What it tests
Two-set overlap	1	~2	Either/only/neither counting with two groups
Three-set overlap	1	~1-2	At least one/none/only counting with three groups

Prep strategy

First pass: memorise the two-set formula and drill 'either,' 'only,' and 'neither' problems until automatic.
Second pass: memorise the three-set formula, paying specific attention to the triple-overlap add-back step.
Final pass: practice mixed two-set and three-set problems under time pressure to build speed.

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Memorise both the two-set and three-set inclusion-exclusion formulas cold — nearly every question in this topic is a direct application.
For 'none' questions, always compute 'at least one' first via the formula, then subtract from the total.
For diagrammatic category questions, explicitly check whether any category is a genuine subset of another before assuming three separate overlapping circles.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Market research and survey analysis

Computing overlap between customer segments (e.g., people who use both Product A and Product B) is a direct, everyday application of two-set and three-set inclusion-exclusion.

Database query logic

SQL UNION, INTERSECT, and set-difference operations directly mirror the Venn diagram counting logic taught in this chapter.

Where else this topic is tested

Prepare once, score in every exam that asks it.

SSC CGL / CHSL ReasoningHigh overlap — Venn diagram counting problems are a recurring topic across many government exams' reasoning sections

Bank PO / Clerk ReasoningModerate to high overlap in two-set and three-set counting question styles

📝 My notes

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

Subtracting all three pairwise overlaps (A∩B, B∩C, A∩C) removes anyone in all three groups a total of three times (once per pairwise subtraction) when they should only be excluded from the double-counting a net one extra time — adding the triple overlap back once corrects this exactly.

Compute 'at least one' using the inclusion-exclusion formula first, then subtract from the total — this is more reliable than trying to reason directly about the region outside all the circles.