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Logical Reasoningतार्किक क्षमता · वेन आरेख

Venn Diagramsवेन आरेख questions and answersप्रश्न और उत्तर

Venn diagram questions count people who belong to one, two or three overlapping groups: students who like cricket or kabaddi, families who own a car or a scooter. Every such question is solved the same way, by filling the diagram from the middle outward.

  • Unlimitedअसीमित fresh questionsनए प्रश्न
  • Answersउत्तर always one tap awayहमेशा एक टैप दूर
  • 1 subtopicउप-टॉपिक
  • SSC CGL ~2
  • IBPS PO ~1
  • TCS NQT ~1

Practice questions are in English for now.अभ्यास के प्रश्न अभी अंग्रेज़ी में हैं।

How venn diagrams worksवेन आरेख को समझिए

Each figure in the question counts a whole circle or a whole overlap, and that figure already includes the people in the smaller regions inside it. “30 like cricket” includes those who also like kabaddi. That is why you subtract overlaps instead of adding them.

Start with the innermost region. With three groups, write the “all three” number in the centre, subtract it from each two-group figure to get the “exactly these two” regions, then subtract all of those from each circle to get the “only” regions. Whatever is left of the total sits outside every circle.

Inclusion–exclusion is the same idea as a formula: add the circles, subtract the pairwise overlaps (because they were counted twice), and add the triple overlap back (because subtracting three pairs removed it once too often). Use the formula for the union and the diagram for questions about “only” and “exactly”.

Tricks that save timeसमय बचाने वाली ट्रिक्स

All tricks →सभी ट्रिक्स →

Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।

Formulaसूत्रRarely taughtकम सिखाई जाने वाली✓ Checked on 1,000 cases✓ 1,000 मामलों पर जाँची गई

Exactly one = S₁ − 2S₂ + 3T; exactly two = S₂ − 3T

Use it when:कब इस्तेमाल करें: A three-group question asks how many are in exactly one or exactly two groups, and you do not want to draw the full diagram.

  1. S₁ = sum of the three circle totals; S₂ = sum of the three pair totals; T = all three.
  2. Exactly two = S₂ − 3T.
  3. Exactly one = S₁ − 2S₂ + 3T.
Exampleउदाहरण

Circles 20, 25, 30; pairs 8, 6, 5; all three 3. Exactly one?

→ S₁ = 75, S₂ = 19: 75 − 38 + 9 = 46.

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: Someone in exactly two groups appears once in S₂ and someone in all three appears three times, so S₂ − 3T counts the exactly-two people. In S₁ an exactly-two person appears twice and an all-three person three times; subtracting 2S₂ removes them four and six times, so adding 3T back leaves only the exactly-one people.

Limits:सीमाएँ: Same as inclusion–exclusion: the pair totals must include the people in all three.

Formulaसूत्र✓ Checked on 500 cases✓ 500 मामलों पर जाँची गई

Add circles, subtract pairs, add the triple

Use it when:कब इस्तेमाल करें: You need “at least one” or “none” from circle totals and overlap totals.

  1. Two groups: at least one = a + b − both.
  2. Three groups: at least one = (a + b + c) − (ab + bc + ca) + abc.
  3. None = total − at least one.
Exampleउदाहरण

40 like tea, 30 like coffee, 12 like both, 70 people in all. How many like neither?

→ 70 − (40 + 30 − 12) = 12.

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: A person in exactly two groups is counted twice in the circle totals and removed once by the pair total. A person in all three is counted three times, removed three times, then added once. Everyone in at least one group ends up counted exactly once.

Limits:सीमाएँ: The pair figures must be “both A and B” (including those in all three). If a question gives “only A and B” instead, add those regions directly without subtracting the triple.

Solved examplesहल किए गए उदाहरण

Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।

  1. Example 1उदाहरण 1 · Venn diagram counting (two and three sets) · Easyआसान

    Of 63 employees in an office, 43 speak Tamil, 22 speak Telugu and 14 speak both. How many speak exactly one of the two?

    1. 29
    2. 37
    3. 51
    4. 65
    Answer:उत्तर: 37
    1. Only Tamil = 43 − 14 = 29; only Telugu = 22 − 14 = 8.
    2. At least one = 43 + 22 − 14 = 51; neither = 63 − 51 = 12.
    3. Answer: 37.

  2. Example 2उदाहरण 2 · Venn diagram counting (two and three sets) · Mediumमध्यम

    Of 71 readers surveyed in a town, 43 read a Marathi paper, 36 read a Hindi paper and 7 read neither. How many read only a Marathi paper?

    1. 21
    2. 28
    3. 35
    4. 43
    Answer:उत्तर: 28
    1. At least one = 71 − 7 = 64.
    2. Both = 43 + 36 − 64 = 15.
    3. Only a Marathi paper = 43 − 15 = 28; only a Hindi paper = 36 − 15 = 21.
    4. Answer: 28.

  3. Example 3उदाहरण 3 · Venn diagram counting (two and three sets) · Hardकठिन

    Of 66 employees in an office, 25 speak Tamil, 33 speak Telugu and 22 speak Kannada; 7 speak both Tamil and Telugu, 8 speak both Telugu and Kannada, 4 speak both Tamil and Kannada, and 2 speak all three. How many speak none of the three?

    1. 3
    2. 5
    3. 7
    4. 63
    Answer:उत्तर: 3
    1. All three: 2. Exactly two: Tamil–Telugu 7 − 2 = 5, Telugu–Kannada 8 − 2 = 6, Tamil–Kannada 4 − 2 = 2.
    2. Only Tamil = 25 − 5 − 2 − 2 = 16; only Telugu = 20; only Kannada = 12.
    3. At least one = 80 − 19 + 2 = 63; none = 66 − 63 = 3.
    4. Answer: 3.

  4. Example 4उदाहरण 4 · Venn diagram counting (two and three sets) · Mediumमध्यम

    Of 31 shoppers at a mela, 15 bought bangles, 20 bought sweets and 6 bought neither. How many bought both?

    1. 4
    2. 10
    3. 16
    4. 25
    Answer:उत्तर: 10
    1. At least one = 31 − 6 = 25.
    2. Both = 15 + 20 − 25 = 10.
    3. Only bangles = 15 − 10 = 5; only sweets = 20 − 10 = 10.
    4. Answer: 10.

  5. Example 5उदाहरण 5 · Venn diagram counting (two and three sets) · Mediumमध्यम

    Of 74 readers surveyed in a town, 46 read a Marathi paper, 42 read a Hindi paper and 5 read neither. How many read only a Marathi paper?

    1. 23
    2. 27
    3. 32
    4. 46
    Answer:उत्तर: 27
    1. At least one = 74 − 5 = 69.
    2. Both = 46 + 42 − 69 = 19.
    3. Only a Marathi paper = 46 − 19 = 27; only a Hindi paper = 42 − 19 = 23.
    4. Answer: 27.

  6. Example 6उदाहरण 6 · Venn diagram counting (two and three sets) · Mediumमध्यम

    Of 30 readers surveyed in a town, 10 read a Marathi paper, 8 read a Hindi paper and 15 read neither. How many read both?

    1. 3
    2. 4
    3. 15
    4. 18
    Answer:उत्तर: 3
    1. At least one = 30 − 15 = 15.
    2. Both = 10 + 8 − 15 = 3.
    3. Only a Marathi paper = 10 − 3 = 7; only a Hindi paper = 8 − 3 = 5.
    4. Answer: 3.

Common mistakesआम गलतियाँ

  • Adding the two circle totals without subtracting the overlap, which counts the “both” group twice.
  • Subtracting the pair overlaps but forgetting to add the triple overlap back.
  • Answering “at least one” when the question asks for “exactly one”.

Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न

How do I find how many like neither?

Find the union first: n(A) + n(B) − n(both). Then subtract it from the total.

Why is the triple overlap added back?

People in all three groups are counted three times in the circles and removed three times with the pairs, so they would vanish. Adding the triple once counts them exactly once.

How do I count “exactly two”?

Take each pairwise figure, subtract the people in all three, and add the three results: S₂ − 3T.

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