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JEE Main & Advanced Mathematics Sets Question Bank Self Evaluation Test - Sets

  • question_answer
    A market research group conducted a survey of 2000 consumers and reported that 1720 consumers like product \[{{P}_{1}}\] and 1450 consumers like product \[{{P}_{2}}.\] What is the least number that must have liked both the products?

    A) 1150                

    B) 2000

    C) 1170    

    D) 2500

    Correct Answer: C

    Solution :

    [c]Let U be the set of all consumers who were questioned, A be the set of consumers who liked product \[{{P}_{1}}\] and B be the set of consumers who liked product \[{{P}_{2}}\].
    It is given that \[n(U)=2000,n(A)=1720,n(B)\]
    \[=\,\,1450,\,\,n(A\cup B)=n(A)+n(B)-n(A\cap B)\]
    \[=1720+1450-n(A\cap B)\]
    \[=3170-n(A\cap B)\]
    Since, \[A\cup B\subseteq U\therefore n(A\cup B)\le n(U)\]
    \[\Rightarrow 3170-n(A\cap B)\le 2000\]
    \[\Rightarrow 3170-2000\le n(A\cap B)\]
    \[\Rightarrow n(A\cap B)\ge 1170\]
    Thus, the least value of \[n(A\cap B)\]is 1170.
    Hence, the least number of consumers who liked both the products is 1170.


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