Venn Diagram – MBA CET questions with solutions

Approach: Venn diagram classification — check which pairs overlap and which are disjoint.

Solution:
Passengers and Airplane: some passengers travel by airplane → overlap.
Passengers and Taxi: some passengers travel by taxi → overlap.
Airplane and Taxi: two different vehicles, no common member → disjoint.
So Passengers is a middle circle overlapping both, while the end circles do not touch: three circles in a chain.
Figure D shows exactly this.

Why the other options are wrong:
A: three separate circles — ignores the passenger overlaps.
B: three mutually overlapping circles — would need something that is both airplane and taxi.
C: concentric circles — would mean one class wholly contains another.
E: two disjoint sets inside one big set — would mean all airplanes and taxis are passengers.

Cetking shortcut: Two unrelated items + one item linked to both = chain diagram.

Answer: D

Approach: Venn diagram region reading — inside the required shapes, outside the excluded one.

Solution:
Men educated and employed but not married = inside Triangle AND Rectangle AND Circle, but outside Square.
Regions in Triangle + Rectangle + Circle: 5 and 9.
Region 9 is also inside the Square (married); region 5 lies above the square's top edge.
So the required region is 5.

Why the other options are wrong:
A: 3 is triangle + rectangle only — not employed.
B: 9 is in all four shapes — married.
D: 8 is triangle + rectangle + square — married and not employed.
E: 11 is circle + square only — neither male nor educated.

Cetking shortcut: List the regions common to the three 'yes' shapes first (5, 9), then strike the one inside the 'no' shape.

Answer: C

Approach: Match the diagram: one class inside another + one completely separate class.

Solution:
Diagram: a small circle wholly inside a bigger circle, and a third circle outside both.
A: every Thief is a Criminal (Thief inside Criminal); a Judge is neither, so Judge is separate. ✓
So Judge, Thief, Criminal fits the diagram.

Why the other options are wrong:
B: Father and Mother are two separate classes both inside Parents.
C: Teacher, College, Students are different kinds of things with no subset pattern matching.
D: Squares ⊂ Rectangles ⊂ Polygons — three nested circles, no separate one.
E: Birds ⊂ Animals ⊂ Living Things — again three nested circles.

Cetking shortcut: Look for exactly one subset pair plus one unrelated item — only A has an outsider.

Answer: A

Approach: Group counting — add every group containing the language.

Solution:
Stated groups: 10 + 5 + 10 + 12 + 13 + 5 = 55.
Remaining (Bengali and Hindi) = 60 − 55 = 5.
Bengali speakers = only Bengali 10 + Bengali & Hindi 5 = 15.
Marathi speakers = Marathi & Hindi 12 + English, Tamil & Marathi 13 = 25.
Ratio = 15 : 25 = 3/5.

Why the other options are wrong:
A: 15/32 adds the wrong groups to Marathi.
B, C: wrong group totals.
E: 5/3 is the ratio inverted.

Cetking shortcut: Find the 'remaining' group first — it is the only missing number.

Answer: D

Approach: Venn diagram: read the relationship between the circles, not their sizes.

Solution:
The figure shows three circles (one larger, two smaller) with no overlap and none inside another ⇒ three completely unrelated sets.
Anxiety, Intelligence and Strength are three different, unrelated qualities: no one is a part of another and none is a type of another ⇒ three separate circles.
Circle size in such questions does not show a 'bigger class'; only enclosure or overlap shows a relation.

Why the other options are wrong:
A: Potato and Cabbage are vegetables ⇒ two small circles inside a big one.
B: Males, Females, Hermaphrodites are all inside Animals ⇒ nested.
C: Table and Chair are inside Furniture ⇒ nested.
D: Districts inside States inside Country ⇒ concentric circles.

Cetking shortcut: Separate circles = 'no relation at all'. Any option with a 'type of' link (vegetable, furniture, animal, country) needs enclosure and is out.

Answer: E

Approach: Venn diagram from statements: draw pairwise relations.

Solution:
Some cricket players are tennis players ⇒ Cricket and Tennis overlap.
Some tennis players are hockey players ⇒ Tennis and Hockey overlap.
No cricket player is a hockey player ⇒ Cricket and Hockey do not touch.
So Tennis is the middle circle overlapping both sides, while Cricket and Hockey stay apart ⇒ a chain of three circles in a row: Figure C.

Why the other options are wrong:
A: all three circles overlap ⇒ some cricket players would be hockey players.
B: one circle is separate from the overlapping pair ⇒ one 'some' relation missing.
D: one circle sits inside another ⇒ suggests 'all', not 'some'.
E: only two circles.

Cetking shortcut: 'A some B, B some C, A no C' is always the three-circle chain with B in the middle.

Answer: C

Approach: Rectangle = wheat, square = gram, circle = maize, triangle = rice; check which shapes enclose each region.

Solution:
Region map: 1 = rice only; 2 = maize + rice; 3 = maize only; 4 = wheat + maize; 5 = wheat only; 6 = wheat + maize + gram; 7 = all four; 8 = wheat + maize + rice; 9 = maize + gram + rice; 10 = maize + gram; 11 = gram only.
Region 7 lies inside the rectangle, the square, the circle and the triangle → 7.

Why the other options are wrong:
B 8: above the square (no gram).
C 9: right of the rectangle edge (no wheat).
D 2: maize + rice only.
E 1: rice only.

Cetking shortcut: Look for the region that is inside the square AND left of the rectangle's right edge AND above the triangle's base.

Answer: A

Approach: Rectangle = wheat, square = gram, circle = maize, triangle = rice; check which shapes enclose each region.

Solution:
Region map: 1 = rice only; 2 = maize + rice; 3 = maize only; 4 = wheat + maize; 5 = wheat only; 6 = wheat + maize + gram; 7 = all four; 8 = wheat + maize + rice; 9 = maize + gram + rice; 10 = maize + gram; 11 = gram only.
Region 4 is inside the rectangle and the circle only → 4.

Why the other options are wrong:
A 8: also rice.
B 6: also gram.
C 5: wheat only.
E 3: maize only.

Answer: D

Approach: Rectangle = wheat, square = gram, circle = maize, triangle = rice; check which shapes enclose each region.

Solution:
Region map: 1 = rice only; 2 = maize + rice; 3 = maize only; 4 = wheat + maize; 5 = wheat only; 6 = wheat + maize + gram; 7 = all four; 8 = wheat + maize + rice; 9 = maize + gram + rice; 10 = maize + gram; 11 = gram only.
Region 1 is the tip of the triangle outside every other shape → 1.

Why the other options are wrong:
A 5: wheat only.
C 2: rice + maize.
D 11: gram only.
E 3: maize only.

Answer: B

Approach: Rectangle = wheat, square = gram, circle = maize, triangle = rice; check which shapes enclose each region.

Solution:
Region map: 1 = rice only; 2 = maize + rice; 3 = maize only; 4 = wheat + maize; 5 = wheat only; 6 = wheat + maize + gram; 7 = all four; 8 = wheat + maize + rice; 9 = maize + gram + rice; 10 = maize + gram; 11 = gram only.
Region 3 is inside the circle only → 3.

Why the other options are wrong:
A 10: maize + gram.
B 2: maize + rice.
D 4: maize + wheat.
E 11: gram only.

Answer: C

Approach: Rectangle = wheat, square = gram, circle = maize, triangle = rice; check which shapes enclose each region.

Solution:
Region map: 1 = rice only; 2 = maize + rice; 3 = maize only; 4 = wheat + maize; 5 = wheat only; 6 = wheat + maize + gram; 7 = all four; 8 = wheat + maize + rice; 9 = maize + gram + rice; 10 = maize + gram; 11 = gram only.
Region 2 is inside the circle and the triangle only → 2.

Why the other options are wrong:
A 9: also gram.
B 8: also wheat.
D 7: all four.
E 5: wheat only.

Answer: C

Approach: Work in 'passed' regions; failed only in SS = passed in Maths and Science only.

Setting up (common to the set): Total 450. Pass all 3 = 167, fail all 3 = 60. Pass only SS = 62, only Maths = 48, only Science = 52. Let x = pass SS & Maths only, y = pass SS & Sci only, z = pass Maths & Sci only.
Failed SS (175) = 60 + 48 + 52 + z → z = 15.
Failed Maths (199) = 60 + 62 + 52 + y → y = 25.
Failed Science (191) = 60 + 62 + 48 + x → x = 21.
Check: 167 + 60 + 62 + 48 + 52 + 21 + 25 + 15 = 450 ✓

Solution: Failed in Social Sciences only = passed Maths & Science only = z = 15.

Why the other options are wrong:
B 21 = x (failed Science only).
C 30, D 42: no region equals these.
E 199: total failed in Maths.

Cetking shortcut: Failed in X = fail-all + 'passed only' in the other two + passed both others → one subtraction gives each 'exactly two passed' region.

Answer: A

Approach: Failed exactly one ⇔ passed exactly two.

Setting up (common to the set): Total 450. Pass all 3 = 167, fail all 3 = 60. Pass only SS = 62, only Maths = 48, only Science = 52. Let x = pass SS & Maths only, y = pass SS & Sci only, z = pass Maths & Sci only.
Failed SS (175) = 60 + 48 + 52 + z → z = 15.
Failed Maths (199) = 60 + 62 + 52 + y → y = 25.
Failed Science (191) = 60 + 62 + 48 + x → x = 21.
Check: 167 + 60 + 62 + 48 + 52 + 21 + 25 + 15 = 450 ✓

Solution: x + y + z = 21 + 25 + 15 = 61.
(Also 450 − 167 − 60 − 162 = 61.)

Why the other options are wrong:
A 152 / B 144 / D 56: partial sums or wrong regions.
E 191: total failed in Science.

Answer: C

Approach: Add the two 'Maths + one other' regions and the 'all three' region.

Setting up (common to the set): Total 450. Pass all 3 = 167, fail all 3 = 60. Pass only SS = 62, only Maths = 48, only Science = 52. Let x = pass SS & Maths only, y = pass SS & Sci only, z = pass Maths & Sci only.
Failed SS (175) = 60 + 48 + 52 + z → z = 15.
Failed Maths (199) = 60 + 62 + 52 + y → y = 25.
Failed Science (191) = 60 + 62 + 48 + x → x = 21.
Check: 167 + 60 + 62 + 48 + 52 + 21 + 25 + 15 = 450 ✓

Solution: x + z + 167 = 21 + 15 + 167 = 203.
Check: total passed Maths = 450 − 199 = 251; minus Maths-only 48 = 203 ✓

Why the other options are wrong:
A 210: adds y instead of z.
C 170, D 94: partial sums.
E 52: Science only.

Cetking shortcut: (Total − failed in Maths) − passed Maths only = 251 − 48 = 203.

Answer: B

Approach: Failed exactly two ⇔ passed exactly one.

Setting up (common to the set): Total 450. Pass all 3 = 167, fail all 3 = 60. Pass only SS = 62, only Maths = 48, only Science = 52. Let x = pass SS & Maths only, y = pass SS & Sci only, z = pass Maths & Sci only.
Failed SS (175) = 60 + 48 + 52 + z → z = 15.
Failed Maths (199) = 60 + 62 + 52 + y → y = 25.
Failed Science (191) = 60 + 62 + 48 + x → x = 21.
Check: 167 + 60 + 62 + 48 + 52 + 21 + 25 + 15 = 450 ✓

Solution: 62 + 48 + 52 = 162.

Why the other options are wrong:
A 56 / C 152 / E 48: partial sums or a single region.
B 61: failed in exactly one subject.

Answer: D

Approach: 'At least one' is the complement of 'none'.

Solution: 450 − 60 (failed all) = 390.

Why the other options are wrong:
A 450: includes those who failed all.
C 304: wrong sum.
D 167: passed all three.
E 60: failed all three.

Answer: B

Approach: Find the region inside Circle (Workers) and Square (Honest) but outside Triangle (Labours).

Solution: Workers ∩ Honest, not Labours = 45.

Why the other options are wrong:
A 56: honest only.
B 101: 45 + 56 (adds non-workers).
C 23: workers only.
E 11: in all three (they are labours).

Answer: D

Approach: Subtract the whole Design circle from the total.

Solution:
Design circle = 64 (both) + 210 (Design only) = 274.
Not Design = 500 − 274 = 226.
(= 85 Atoms-only + 141 in neither.)

Why the other options are wrong:
A 500: the total.
B 295: 85 + 210.
D 210: Design only.
E 149: the Atoms Thermodynamics circle.

Answer: C

Approach: Find every piece of the Rectangle (Hockey) that lies outside both the Triangle (Tennis) and the Circle (Chess).

Solution: Hockey-only pieces: 5 (left) + 20 (right) = 25.

Why the other options are wrong:
A 5 / B 20: only one of the two pieces.
C 28 / D 23: add or drop a shared region by mistake.

Answer: E

Approach: Pick the region inside Authors and Doctors but outside Females.

Solution:
Triangle = Authors, Rectangle = Doctors, Circle = Females.
Male Doctors who are Authors → inside triangle ∩ rectangle, outside circle.
That region holds 10.
(8 is in all three — female author-doctors; 13 is female doctors who are not authors.)

Why the other options are wrong:
C) 8 — those are females.
A), B), D) — no such region value.

Answer: E

Approach: Region inside Triangle, Rectangle and Circle but outside the Square.

Solution:
Men (triangle) ∩ Educated (rectangle) ∩ Employed (circle), not Married (square).
Region 5 lies in the triangle, rectangle and circle and above the square → 5.
(Region 9 is the same set but inside the square — married.)

Why the other options are wrong:
C) 9 — married.
B) 10 — lies outside the triangle (not men).
A) 7 — not a man and not employed (outside the triangle and the circle).
E) 4 — employed only, not men or educated.

Answer: D

Approach: Take each group as 100% and remove the overlap shares shown for it.

Solution:
Only girls = 100 − 12 − 16 = 72
Only students = 100 − 12 − 14 = 74
Only dancers = 100 − 14 − 16 = 70
→ 72, 74, 70

Answer: A

Approach: Add every region common to Telugu and Tamil.

Solution:
Telugu ∩ Tamil (only these two) = 12
Telugu ∩ Tamil ∩ English = 9
Speak Tamil and Telugu both = 12 + 9 = 21

Why the other options are wrong:
A) 9 — only the all-three region.
B) 12 — misses the 9 who also speak English.
D) 13 — Telugu–English region.
E) 40 — wrong addition.

Answer: C

Approach: Regions inside the Rectangle (MBAs) but outside the Triangle (Musicians).

Solution:
The rectangle contains regions 11, 2, 3, 4 and 5.
Regions 2, 3, 4 and 5 all lie inside the triangle (musicians).
Only region 11 is outside the triangle → 11 MBAs are not musicians.

Why the other options are wrong:
A) 10 — professionals only, not MBAs.
B) 7 — not inside the rectangle.
C) 12 matches no region; E) 13 = 11 + 2 wrongly adds a musician region.

Answer: D

Approach: Read which circles contain the region and which do not.

Solution:
Region 1 lies inside the Honest (yellow), Advocates (green) and Swimmers (olive) circles.
It lies just left of the Europeans (purple) circle — outside it (region 2 is the one inside all four).
So region 1 = honest advocates who are swimmers and NOT European.

Why the other options are wrong:
A: says European and omits advocates.
B: does not exclude Europeans (that covers regions 1 and 2).
D: omits advocates.
E: European advocates who are honest swimmers = region 2.

Cetking shortcut: For any region, list IN-circles and OUT-circles; the right option names both.

Answer: C

Approach: Two-set inclusion–exclusion.

Solution:
Cricket = 50% of 200 = 100; Football = 20% = 40; Both = 10% = 20.
At least one = 100 + 40 − 20 = 120.
Neither = 200 − 120 = 80.

Cetking shortcut: Work in %: 50 + 20 − 10 = 60% play some game → 40% of 200 = 80.

Answer: C

Approach: Read the Venn figure: Tennis (triangle) regions that lie outside the Hockey rectangle.

Solution:
Tennis regions outside the rectangle: top tip 5, Tennis ∩ Chess above the rectangle 11, Tennis ∩ Chess below the rectangle 14, bottom tip 5.
Tennis not Hockey = 5 + 11 + 14 + 5 = 35.
(10 lies in all three shapes, so it is excluded.)

Why the other options are wrong:

• B 56 / E 45 – include hockey regions (10 etc.).
• C 37 / D 29 – miss or wrongly add a region.

Cetking shortcut: Shade the triangle, then strike out everything inside the rectangle – add what is left.

Answer: A

Approach: Three-set union with 'at least two' and 'all three'.

Solution:
Sum of individual counts = 15 + 15 + 20 = 50.
At least two = 8, all three = 2 → exactly two = 8 − 2 = 6.
Union = Σ − (exactly two) − 2 × (all three) = 50 − 6 − 2 × 2 = 40.

Cetking shortcut: Σ counts exactly-two buyers twice and all-three buyers thrice – subtract the extras: 6 × 1 + 2 × 2 = 10.

Answer: D

Approach: Read the region inside only the triangle.

Solution:
Triangle (Science) regions: 2 (outside circle and rectangle), 6 (with Maths), 3 (with Maths and History).
Science only = region in the triangle alone = 2.

Why the other options are wrong:

• A 6 – Maths and Science.
• C 3 – all three subjects (or History only).
• D 7 – Maths and History.
• E 4 – Maths only.

Answer: B

Approach: Add every region that contains football.

Solution:
Football only = 20; Football & Hockey = 30; Football & Cricket = 5.
Football lovers = 20 + 30 + 5 = 55.
(Check: only basketball = 100 − 20 − 30 − 15 − 25 − 5 = 5.)

Why the other options are wrong:

• B 20 – football only.
• A 35 / C 25 – miss one overlap.

Answer: D

Approach: Find the region inside the triangle and the circle but outside the rectangle.

Solution:
Triangle = Women, Circle = Lawyers, Rectangle = Doctors.
Region 3 lies inside the triangle and the circle, above the rectangle's top edge → women lawyers who are not doctors.

Why the other options are wrong:
10 – inside all three (women lawyers who are doctors).
12 – triangle and rectangle, outside the circle (women doctors, not lawyers).
8 – circle and rectangle only (male lawyer-doctors).
2 – not a region in the figure.

Answer: B

Approach: Three mutually overlapping circles need every pair and all three together to be able to overlap.

Solution:
Extroverts, Women, Tall: a person can be any combination — an extrovert, a woman and tall, or any two, or one — so all three sets overlap, exactly as drawn.

Why the other options are wrong:
A – Pens are inside Stationery; Pigeons are separate.
B – Mothers (female) and Sons-in-law (male) cannot overlap.
C – Men and Women do not overlap.
D – Balls, Wickets, Aliens are three separate things.

Answer: E

Approach: Pick the region inside circle, square and rectangle but outside the triangle.

Solution:
Circle = intelligent, Square = hard working, Rectangle (tilted) = loyal, Triangle = post graduate.
Region 4 lies inside the circle, inside the square and inside the tilted rectangle, and below the triangle's base → intelligent, hard working, loyal, not post graduate.

Why the other options are wrong:
5 – circle and rectangle but outside the square (not hard working).
6 – circle only.
9 – no such region in the figure.
11 – inside the triangle (post graduate).

Answer: B

Approach: Venn diagram probability — favourable region ÷ total of all regions.

Solution:
Regions from the figure: Running only 13, Cycling only 15, Swimming only 14, Running∩Cycling only 4, Running∩Swimming only 10, Cycling∩Swimming only 7, all three 5.
Total people = 13 + 15 + 14 + 4 + 10 + 7 + 5 = 68.
Likes Running AND Cycling = (R∩C only) + (all three) = 4 + 5 = 9.
Probability = 9/68, i.e. 9 : 68 as written in the options.

Why the other options are wrong:
A (2 : 5), B (1 : 3), D (1 : 20), E (4 : 13): none has denominator 68 — the total of the diagram.
Taking only the '4' region (4/68 = 1 : 17) forgets the centre region, which also likes both.

Cetking shortcut: The denominator must be the full total (68); only option C has it.

Answer: C

Approach: Two-set Venn: Total = only V + only H + both (everyone plays at least one).

Solution:
Violin total = 30, both = 20 → only violin = 30 − 20 = 10.
Only harmonium = 80 − 10 − 20 = 50.
Harmonium total = 50 + 20 = 70.
So: 70, 50 and 10.

Cetking shortcut: Only option A ends with 10 for only-violin (30 − 20) — the rest can be eliminated instantly.

Answer: A

Approach: Read the exact Venn region.

Solution:
Chess ∩ Ludo regions: 4 (outside Carrom) and 3 (inside Carrom).
'Chess and Ludo but not Carrom' = the region inside Chess and Ludo but outside Carrom (and outside Table Tennis) = 4.
The total of 600 is not needed.

Why the other options are wrong:
A: 3 is Chess ∩ Ludo ∩ Carrom.
C: 10 is Table Tennis ∩ Chess (or Chess ∩ Carrom).
D: 8 is Carrom ∩ Ludo only.
E: 5 is TT ∩ Chess ∩ Carrom.

Answer: B

Approach: Three-set inclusion–exclusion, everyone in at least one set.

Solution: Total = F + T + C − (F∩T + T∩C + F∩C) + (F∩T∩C)
256 = 144 + 123 + 132 − (58 + 25 + 63) + x = 399 − 146 + x = 253 + x → x = 3.
Only tennis = T − F∩T − T∩C + x = 123 − 58 − 25 + 3 = 43.

Cetking shortcut: Only-T = T − (pairs touching T) + all-three; find x from the total first.

Answer: B

Approach: Venn figure — read the overlap of the two named shapes.

Solution: Artists = rectangle, players = circle.
Regions inside both the circle and the rectangle: 22 (artist-players, not doctors) and 3 (artist-players who are also doctors).
Artists who are players = 22 + 3 = 25.

Why the other options are wrong:
A 32 — adds wrong regions.
B 15 — no such combination.
D 17 — triangle-only (doctors only) region.
E 12 — no such combination.

Cetking shortcut: Include every region inside both shapes, whatever third shape also covers it — the 3 sitting under the triangle line is the usual miss.

Answer: C

Approach: Venn regions — (graduates OR teachers) AND NOT politicians.

Solution: Reading the figure (big triangle = politicians, small triangle = teachers, circle = graduates, rectangle = MPs):
E = teacher only; D = teacher + politician; C = teacher + graduate + politician; F = teacher + graduate + politician + MP;
B = graduate + politician; G = graduate + politician + MP; A = graduate only; H = politician + MP; L = politician only.
Regions outside the big triangle that lie in the small triangle or the circle: E (teacher only) and A (graduate only).

Why the other options are wrong:
A B, G — inside the politicians' triangle.
B G, H — inside the politicians' triangle (H is not a graduate or teacher either).
D E, F — F is a politician.
E A, C — C is a politician.

Cetking shortcut: Only two letters lie outside the big triangle and inside another shape — E and A.

Answer: C

Approach: Venn regions — graduates AND politicians AND NOT MPs.

Solution: Reading the figure (big triangle = politicians, small triangle = teachers, circle = graduates, rectangle = MPs):
E = teacher only; D = teacher + politician; C = teacher + graduate + politician; F = teacher + graduate + politician + MP;
B = graduate + politician; G = graduate + politician + MP; A = graduate only; H = politician + MP; L = politician only.
Graduate politicians (circle ∩ big triangle) = B, C, F, G.
Remove those inside the rectangle (MPs): F and G.
Left: B, C.

Why the other options are wrong:
B L, B — L is not a graduate.
C D, H — neither is a graduate.
D D, L — neither is a graduate.
E L, H — neither is a graduate; H is an MP.

Cetking shortcut: Every option except A contains a non-graduate letter (L, D or H) — eliminate on sight.

Answer: A

Approach: Venn regions — politicians AND NOT graduates AND NOT teachers.

Solution: Reading the figure (big triangle = politicians, small triangle = teachers, circle = graduates, rectangle = MPs):
E = teacher only; D = teacher + politician; C = teacher + graduate + politician; F = teacher + graduate + politician + MP;
B = graduate + politician; G = graduate + politician + MP; A = graduate only; H = politician + MP; L = politician only.
Inside the big triangle but outside both the circle and the small triangle: L (politician only) and H (politician + MP).
Being an MP does not matter for this condition.

Why the other options are wrong:
A E, F — E is not a politician; F is a teacher and graduate.
B D, E — D is a teacher; E is not a politician.
D C, D — both are teachers.
E C, L — C is a teacher and graduate.

Answer: C

Approach: Venn regions — MPs AND graduates AND teachers.

Solution: Reading the figure (big triangle = politicians, small triangle = teachers, circle = graduates, rectangle = MPs):
E = teacher only; D = teacher + politician; C = teacher + graduate + politician; F = teacher + graduate + politician + MP;
B = graduate + politician; G = graduate + politician + MP; A = graduate only; H = politician + MP; L = politician only.
The only region common to the rectangle, the circle and the small triangle is F.

Why the other options are wrong:
A G — not a teacher.
B H — not a graduate or teacher.
D C — not an MP.
E B — not an MP or teacher.

Answer: C

Approach: Three-set Venn diagram — 'exactly one group' = sum of the 'only' regions.

Solution:
Only Art = 6, only Chess = 2, only Drama = 3.
All three = 5; Chess & Art only = 2.
Check with Art total 15: 6 + 5 + 2 + (Drama & Art only) = 15 → Drama & Art only = 2.
Check with total 30: 6 + 2 + 3 + 5 + 2 + 2 + (Chess & Drama only) = 30 → Chess & Drama only = 10. Data is consistent.
Exactly one group = 6 + 2 + 3 = 11.

Why the other options are wrong:
B (9), D (14), E (13): add or drop one of the 'only' regions.
C (15): is the Art total, not 'exactly one'.

Cetking shortcut: 'Exactly one' needs only the three 'only' numbers — they are given directly.

Answer: A

Approach: Venn diagram: take the union of two circles and remove everything inside the third.

Solution:
Volleyball circle = A, B, C, D; Football circle = D, C, F, G; Basketball circle = B, C, E, F.
Volleyball or Football (or both) = A, B, C, D, F, G.
Remove every region inside Basketball (B, C, F): left with A, D, G.
Required region = A + D + G.

Why the other options are wrong:
A (A+G): misses D, students who play both Volleyball and Football but not Basketball.
C (C+D) and D (A+C+D): include C, which lies inside Basketball, and miss G.
E (A+G+D) lists the same three regions as B; B is the keyed option.

Cetking shortcut: Shade the "NOT" circle first, then read the unshaded parts of the other two circles.

Answer: B

Approach: Venn diagram: find the region common to all four shapes.

Solution:
Rectangle = wheat, square = gram, circle = maize, triangle = rice. Read each numbered region by asking: inside which shapes is it?
Region 7 lies inside the rectangle, the square, the circle and the triangle at the same time.
So all four crops are grown in region 7.

Why the other options are wrong:
B 8: inside rectangle, circle, triangle but above the square — gram missing.
C 9: inside square, circle, triangle but right of the rectangle — wheat missing.
D 2: circle and triangle only.
E 1: triangle only.

Cetking shortcut: The all-shape region is always the innermost cell where the most boundaries cross — here the small cell 7.

Answer: A

Approach: Venn diagram: region inside exactly two named shapes.

Solution:
Rectangle = wheat, square = gram, circle = maize, triangle = rice. Read each numbered region by asking: inside which shapes is it?
Wheat and maize only = inside rectangle and circle, outside square and triangle.
Region 4 is in the rectangle and circle, above the square and left of the triangle → 4.

Why the other options are wrong:
A 8: also inside the triangle (rice).
B 6: also inside the square (gram).
C 5: rectangle only.
E 3: circle only.

Answer: D

Approach: Venn diagram: region inside one shape only.

Solution:
Rectangle = wheat, square = gram, circle = maize, triangle = rice. Read each numbered region by asking: inside which shapes is it?
Rice only = inside the triangle and outside rectangle, square and circle.
Region 1 (the tip of the triangle outside the circle) → 1.

Why the other options are wrong:
A 5: rectangle only (wheat).
C 2: triangle and circle.
D 11: square only (gram).
E 7: all four.

Answer: B

Approach: Venn diagram: region inside one shape only.

Solution:
Rectangle = wheat, square = gram, circle = maize, triangle = rice. Read each numbered region by asking: inside which shapes is it?
Maize only = inside the circle, outside rectangle, square and triangle.
Region 3 (top of the circle) → 3.

Why the other options are wrong:
A 10: also inside the square (gram).
B 2: circle and triangle.
D 4: circle and rectangle.
E 11: square only.

Answer: C

Approach: Two-set Venn diagram: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Solution:
TV = 65%, newspaper = 40%, both = 25%.
At least one = 65 + 40 − 25 = 80%.
Neither = 100 − 80 = 20%.

Why the other options are wrong:
A (5%): this is 105 − 100 with the overlap left out.
E (25%): this is the 'both' value.
B (10%), C (15%): these are arithmetic slips.

Cetking shortcut: Neither = 100 − (A + B − both) = 100 − 80 = 20.

Answer: D

Approach: Two-set Venn diagram: n(F ∪ E) = n(F) + n(E) − n(F ∩ E).

Solution:
People who read at least one = 15 − 3 = 12.
12 = 7 + 8 − both, so both = 3.

Why the other options are wrong:
A (0): this assumes no overlap, but 7 + 8 = 15 > 12.
C, D, E: these do not fit 7 + 8 − x = 12.

Cetking shortcut: Both = (F + E) − (Total − Neither) = 15 − 12 = 3.

Answer: B