6.3 Permutations (Order Matters)

Word: D, A, U, G, H, T, E, R (8 distinct letters). Vowels: A, U, E (3). Consonants: D, G, H, T, R (5).

(i) Together: Treat {A, U, E} as one block 'X'. Objects to arrange: {D, G, H, T, R, X} = 6 objects. Arrangements = $6! = 720$. Internal arrangement of vowels = $3! = 6$. Total = $720 \times 6 = 4320$.

(ii) Never Together: Total arrangements = $8! = 40,320$. Never Together = Total - Together = $40,320 - 4,320 = 36,000$.