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    Question

    In how many distinct ways can the letters of the word

    "MANAGEMENT" be arranged so that all the vowels are always together? (Assume identical letters are indistinguishable.)
    A 7560 Correct Answer Incorrect Answer
    B 3350 Correct Answer Incorrect Answer
    C 9520 Correct Answer Incorrect Answer
    D 4520 Correct Answer Incorrect Answer

    Solution

    ATQ, Word: M A N A G E M E N T Counts: A(2), E(2), M(2), N(2), G(1), T(1). Vowels: A, A, E, E (4 letters, with 2 A’s and 2 E’s). Treat them as one block [V]. Consonants: M, M, N, N, G, T (6 letters). Total items to arrange = 6 consonants + 1 vowel-block = 7 items: [V], M, M, N, N, G, T. Step 1: Arrange 7 items: Ways = 7! / (2! × 2!) = 5040 / 4 = 1260. Step 2: Arrange 4 vowels within [V]: A, A, E, E. Ways = 4! / (2! × 2!) = 24 / 4 = 6. Total arrangements = 1260 × 6 = 7560.

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