JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Self Evaluation Test - Permutations and Combinatioins

  • question_answer
    Seven people leave their bags outside temple and while returning after worshiping the deity, picked one bag each at random. In how many ways at least one and at most three of them get their correct bags?

    A) \[^{7}{{C}_{3}}.9{{+}^{7}}{{C}_{5}}.44{{+}^{7}}{{C}_{1}}.265\]

    B) \[^{7}{{C}_{6}}.265{{+}^{7}}{{C}_{5}}.9{{+}^{7}}{{C}_{7}}.44\]

    C) \[^{7}{{C}_{5}}.9{{+}^{7}}{{C}_{2}}.44{{+}^{7}}{{C}_{1}}.265\]

    D) None of these

    Correct Answer: A

    Solution :

    [a] No. of ways getting one correct \[{{=}^{7}}{{C}_{6}}6!\left( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+...+\frac{1}{6!} \right){{=}^{7}}{{C}_{1}}.265\] No. of ways getting two correct \[{{=}^{7}}{{C}_{2}}.5!\left( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+...-\frac{1}{5!} \right){{=}^{7}}{{C}_{2}}.44\] No. of ways getting three correct \[{{=}^{7}}{{C}_{3}}.4!\left( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!} \right){{=}^{7}}{{C}_{3}}.9\] Required no. of ways \[{{=}^{7}}{{C}_{3}}.9{{+}^{7}}{{C}_{2}}.44{{+}^{7}}{{C}_{1}}.265.\]

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