(i) The number of ways to arrange 8 people in 12 seats with no restrictions is given by the permutation formula:

\(^{12}P_8 = \frac{12!}{(12-8)!} = 19,958,400\).

(ii) To ensure Mary and Frances do not sit next to each other, first calculate the total arrangements if they sit together. Treat Mary and Frances as a single unit, so there are 7 units to arrange in 11 seats:

\(^{11}P_7 = \frac{11!}{(11-7)!} = 1,663,200\).

Since Mary and Frances can switch places, multiply by 2:

\(1,663,200 \times 2 = 3,326,400\).

Subtract this from the total unrestricted arrangements:

\(19,958,400 - 3,326,400 = 16,632,000\).

(iii) If all 8 people sit together, treat them as a single block. There are 5 possible positions for this block in 12 seats. The number of ways to arrange the 8 people within the block is:

\(8! = 40,320\).

Multiply by the 5 positions:

\(40,320 \times 5 = 201,600\).