(i) The word AGGREGATE has 9 letters with the following repetitions: G appears 2 times, A appears 2 times, and E appears 2 times. If the first letter is R, we arrange the remaining 8 letters. The number of arrangements is given by:

\(\frac{8!}{2! \times 2! \times 2!} = \frac{40320}{8} = 1680\)

(ii) Treat the 3 G's as a single unit, the 2 A's as a single unit, and the 2 E's as a single unit. This gives us 5 units to arrange: G, A, E, R, T. The number of arrangements is:

\(5! = 120\)

(iii) We need to arrange the letters such that consonants and vowels alternate. There are 5 consonants (G, G, R, T) and 4 vowels (A, A, E, E). Start with a consonant, then alternate:

\(Consonant positions: 5! / (2!) = 60\)

\(Vowel positions: 4! / (2!) = 12\)

Total arrangements: \(5! \times 4! / (2! \times 2!) = 120\)