In how many distinct ways can the letters of the word

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.