Numeric Venn Diagrams: Surveys with Two and Three Groups
Numeric Venn Diagrams: Surveys with Two and Three Groups
Numeric Venn questions give survey counts or percentages with overlaps and ask for "only", "both", "neither" or "exactly two". Two formulas and one habit (fill the innermost region first) solve them all. CBT-2 adds the least and greatest possible overlap.
1. Formula Box
Two sets: n(A or B) = n(A) + n(B) − n(A and B); neither = total − n(A or B)
Only A = n(A) − n(A and B)
Three sets: n(A or B or C) = n(A) + n(B) + n(C) − n(AB) − n(BC) − n(CA) + n(ABC)
Exactly two = (n(AB) + n(BC) + n(CA)) − 3 n(ABC)
Exactly one = union − exactly two − all three
| Bound | Two sets (total N) | Three sets (total N) |
|---|---|---|
| Least overlap | n(A) + n(B) − N (if positive) | All three: n(A) + n(B) + n(C) − 2N (if positive) |
| Greatest overlap | The smaller of n(A), n(B) | The smallest of the three |
✗ 45 drink tea and 38 drink coffee, so 83 drink at least one | ✓ Subtract those counted twice: 45 + 38 − 15 = 68
✗ n(A and B) in three sets already means "A and B only" | ✓ n(AB) includes the people in all three; subtract them for "only"
हिंदी नोट: दो समूहों में "कम से कम एक" = A + B − दोनों। तीन समूहों के प्रश्न में सबसे पहले बीच वाला भाग (तीनों) भरिए, फिर "ठीक दो", फिर "केवल"।
Exam Pointer: Verified NTPC patterns: three-group percentages and counts from a survey (January 2021 CBT-1, January 2017 CBT-2) and residents grouped by the languages they speak (April 2016 CBT-1). Two-set surveys and least or greatest overlaps had no verified NTPC shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Two groups: either, only and neither
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Of 80 people, 45 drink tea, 38 drink coffee and 15 drink both. How many drink neither?
At least one = 45 + 38 − 15 = 68, so neither = 12.
Answer: 12
EXAMATLAS LEVEL
Q. In a town, 40% read paper A, 35% read paper B and 15% read both. 120 people read neither. How many people are there, and how many read only A?
At least one = 40 + 35 − 15 = 60%, so neither = 40% = 120 and the total is 300. Only A = 40 − 15 = 25% = 75.
Answer: 300; 75
Pattern 2: Three groups: all three, exactly one, exactly two
[PYQ: NTPC CBT-1 13-Jan-2021 Shift-1 | NTPC CBT-2 19-Jan-2017 Shift-2]
Survey of 100 people for both questions: 50 like A, 45 like B, 40 like C; 20 like A and B, 15 like B and C, 18 like A and C; 8 like all three.
EXAM LEVEL
Q. How many like none of the three?
Union = 50 + 45 + 40 − 20 − 15 − 18 + 8 = 90, so 10 like none.
Answer: 10
EXAMATLAS LEVEL
Q. How many like exactly one, and how many like exactly two?
Only A = 50 − 20 − 18 + 8 = 20, only B = 45 − 20 − 15 + 8 = 18, only C = 40 − 15 − 18 + 8 = 15: exactly one = 53. Exactly two = (20 − 8) + (15 − 8) + (18 − 8) = 29. Check: 53 + 29 + 8 = 90.
Answer: 53; 29
Pattern 3: Survey grouping and ratios
[PYQ: NTPC CBT-1 27-Apr-2016 Shift-2]
EXAM LEVEL
Q. In a colony of 60 people, 20 speak only Hindi, 15 speak only English and the rest speak both. Find the ratio of Hindi speakers to English speakers.
Both = 60 − 35 = 25. Hindi speakers = 45 and English speakers = 40, a ratio of 9 : 8.
Answer: 9 : 8
EXAMATLAS LEVEL
Q. Of 60 residents, 30 speak Bengali, 25 speak Marathi and 10 speak both. How many speak neither, and what is the ratio of those who speak only Bengali to those who speak only Marathi?
Either = 30 + 25 − 10 = 45, so 15 speak neither. Only Bengali = 20 and only Marathi = 15, a ratio of 4 : 3.
Answer: 15; 4 : 3
CBT-2 LEVEL
Pattern 4: Least and greatest overlap
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. In a group, 70% like tea and 60% like coffee. What are the least and greatest percentages that like both?
Least = 70 + 60 − 100 = 30%. Greatest = the smaller group, 60%.
Answer: 30% and 60%
EXAMATLAS LEVEL
Q. In a class of 50, 40 play cricket, 42 play football and 45 play hockey. What is the least number who play all three?
The numbers not playing each game are 10, 8 and 5, at most 23 students in all. So at least 50 − 23 = 27 play all three. Formula: 40 + 42 + 45 − 2 × 50 = 27.
Answer: 27
60-Second Revision
- Two sets: either = A + B − both; neither = total − either.
- Three sets: fill "all three" first, then "exactly two", then "only".
- Exactly two = sum of pair overlaps − 3 × all three.
- Least overlap of two = A + B − total; of three = sum − 2 × total.
Next Step: Venn Diagrams done. Try the Venn set in the ExamAtlas RRB NTPC 2026 mock tests; label the seven regions before you add a single number.