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Sets and Relations question

2023 · 11 Apr · Shift 1 · Q40
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  5. /2023 · 11 Apr · Shift 1 · Q40

Sets and Relations question

2023 · 11 Apr · Shift 1 · Q40

JEE MainMathematicsSets and RelationsMCQ+4 / −1
An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?
  1. A
    10
  2. B
    15
  3. C
    21
  4. D
    9
View written solutionFree

Correct answer: C

  1. Let the sets of men who got medals in events A,B,CA, B, CA,B,C be A,B,CA, B, CA,B,C respectively.

    Given: n(A)=48,n(B)=25,n(C)=18n(A)=48,\quad n(B)=25,\quad n(C)=18n(A)=48,n(B)=25,n(C)=18 and total number of men who got at least one medal is n(A∪B∪C)=60n(A\cup B\cup C)=60n(A∪B∪C)=60 Also, n(A∩B∩C)=5n(A\cap B\cap C)=5n(A∩B∩C)=5

  2. We use the relation: n(A)+n(B)+n(C)=n(A∪B∪C)+(number in exactly two sets)+2×(number in exactly three sets)n(A)+n(B)+n(C) = n(A\cup B\cup C) + (\text{number in exactly two sets}) + 2\times (\text{number in exactly three sets})n(A)+n(B)+n(C)=n(A∪B∪C)+(number in exactly two sets)+2×(number in exactly three sets)

    Why? Because:

    • a man in exactly one event is counted once,
    • a man in exactly two events is counted twice,
    • a man in exactly three events is counted thrice.

    So if we compare total set counts with union count, the extra counts are:

    • 111 extra for each person in exactly two sets,
    • 222 extra for each person in exactly three sets.

  3. Now substitute the values: 48+25+18=9148+25+18 = 9148+25+18=91

    Hence, 91=60+x+2(5)91 = 60 + x + 2(5)91=60+x+2(5) where xxx is the number of men who received medals in exactly two events.

  4. Solve for xxx: 91=60+x+1091 = 60 + x + 1091=60+x+10 91=70+x91 = 70 + x91=70+x x=21x=21x=21

  5. Therefore, the number of men who received medals in exactly two of the three events is 21\boxed{21}21​

  6. Checking options:

    • A: 101010 ❌
    • B: 151515 ❌
    • C: 212121 ✅
    • D: 999 ❌
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