(i) The word WALLFLOWER consists of 10 letters where L is repeated twice. The number of different arrangements is given by:

\(\frac{10!}{2!} = \frac{3628800}{2} = 302400\)

(ii) To have exactly six letters between the two Ws, consider the positions of the Ws as fixed with six letters in between them. The remaining 8 letters (including the two Ls) can be arranged in the remaining positions. The number of ways to arrange these 8 letters is:

\(\frac{8!}{2!} = \frac{40320}{2} = 20160\)

Since there are three possible positions for the Ws (e.g., W******W*, **W******W, W******W**), we multiply by 3:

\(20160 \times 3 = 20160\)