(i) The word ARGENTINA has 9 letters, with the letter 'A' repeated twice and the letter 'N' repeated twice. The number of different arrangements is given by:

\(\frac{9!}{2!2!} = \frac{362880}{4} = 90720\)

(ii) For arrangements with a consonant at the beginning, then a vowel, and so on alternately, we have 5 positions for consonants and 4 positions for vowels. The number of ways to arrange the consonants is \(5!\) and the number of ways to arrange the vowels is \(4!\). Therefore, the total number of such arrangements is:

\(\frac{5! \times 4!}{2!2!} = \frac{120 \times 24}{4} = 720\)