(i) The word 'GOLD MEDAL' consists of 9 letters where D and L are repeated twice each. The number of different arrangements is given by:

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

(ii) If the two letters D come first and the two letters L come last, we fix these four letters as DD____LL. We need to arrange the remaining 5 letters (G, O, M, E, A) in the middle:

The number of arrangements is given by:

\(5! = 120\)