2024 HCI P2 Q8

A conference hall has \({5}\) doors, labelled A, B, C, D and E, which are located side by side as shown below. The doors are to be painted using \({~4}\) distinct colours, and each door will be painted with a single colour.

1. By considering the number of colours available for each door, find the number of ways to paint the \({5}\) doors such that there is no restriction to the colour of each door.[1]
2. Find the number of ways to paint the doors such that there are no consecutive doors which are of the same colour.[3]
3. Find the number of ways to paint the doors if all \({4}\) colours are to be used.[3]