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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Definition of permutation, Number of permutations with or without repetition, Conditional permutations

  • question_answer
    The number of 4 digit even numbers that can be formed using 0, 1, 2, 3, 4, 5, 6 without repetition is [Kerala (Engg.) 2001]

    A) 120

    B) 300

    C) 420

    D) 20

    Correct Answer: C

    Solution :

    The units place can be filled in 4 ways as any one of 0, 2, 4 or 6 can be placed there. The remaining three places can be filled in with remaining 6 digits in \[^{6}{{P}_{3}}\] = 120 way. So, total number of ways = 4 × 120 = 480. But, this includes those numbers in which 0 is fixed in extreme left place. Numbers of such numbers \[=3\times {{\,}^{5}}{{P}_{2}}\,=3\times 5\times 4=60\]
    0 × × ×
    Fix \[^{5}{{P}_{2}}\] ways 3 ways  (only 2, 4 or 6)
    \ Required number of ways = 480 - 60 = 420.


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