KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    Let X be a set of 5 elements. The number d of ordered pairs (A, B) of subsets of X such that \[A\ne \phi ,\]\[B\ne \phi ,\]\[A\cap B=\phi \]satisfies

    A) \[50\le d\le 100\]

    B) \[101\le d\le 150\]

    C) \[151\le d\le 200\]

    D) \[201\le d\]

    Correct Answer: C

    Solution :

    We have, \[n(X)=5\]
    A and B are subset of X
    Such that \[A\ne \phi ,\]\[B\ne \phi A\cap B=\phi \]
    Total number of order pair (A, B) is \[{}^{5}{{C}_{2}}\times 2!+{}^{5}{{C}_{3}}\left( \frac{3!}{2!}\times 2! \right)+{}^{5}{{C}_{4}}\left( \frac{4!}{3!}2!+\frac{4!}{2!2!} \right)\]\[+{}^{5}{{C}_{5}}\left( \frac{5!}{4!}2!+\frac{5!}{2!3!}\times 2! \right)\]
    \[\therefore \]\[d\in [151,200]\]      \[\Rightarrow \]\[151\le d\le 200\]

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