(i) The total number of arrangements of 9 cards is given by the factorial of 9, which is:

\(9! = 362880\)

(ii) To find the number of arrangements where the pink and green cards are not next to each other, first calculate the total arrangements where they are together. Treat the pink and green cards as a single unit, so there are 8 units to arrange:

\(8! = 40320\)

Within this unit, the pink and green cards can be arranged in 2 ways (PG or GP):

\(8! \times 2 = 80640\)

Subtract this from the total arrangements:

\(362880 - 80640 = 282240\)

(iii) The number of ways to choose and arrange 3 cards from 9 is given by:

\(^9P_3 = \frac{9!}{(9-3)!} = 504\)

(iv) To find the number of arrangements containing the pink card, choose 2 more cards from the remaining 8 and arrange them:

\(^8C_2 \times 3! = 28 \times 6 = 168\)

(v) For arrangements not having the pink card next to the green card, calculate the arrangements where they are together and subtract from the total in part (iii). Treat PG as a single unit, choose 1 more card from the remaining 7, and arrange:

\(7 \times 2 \times 2 = 28\)

Subtract from total arrangements of 3 cards:

\(504 - 28 = 476\)