2022 EJC P2 Q8

In an art installation, three points on the floor, \(X\), \(Y\) and \(Z\), were connected using a rope to form a triangle \(XYZ\). \(9\) bulbs, labelled \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\), \(I\), are then to be placed along the rope to improve the aesthetic appearance.

1. The \(9\) bulbs are first distributed into \(3\) groups, namely Group \(XY\), Group \(YZ\) and Group \(ZX\). Find the number of ways the bulbs can be distributed if each group consists of at least \(2\) bulbs.[2]

In one particular distribution from part (a), the bulbs from Group \(XY\), Group \(YZ\) and Group \(ZX\) are placed on the sides \(XY\), \(YZ\) and \(ZX\)of the triangle \(XYZ\) respectively. The bulbs are arranged as shown in the diagram below.

2. By connecting the bulbs in this arrangement using additional rope, we can form a smaller triangle within the triangle \(XYZ\). For example, the bulbs \(B\), \(I\), \(G\) form a smaller triangle when connected. How many different smaller triangles can be formed?[2]
3. Each bulb can be programmed to produce either red, blue or green light when it is switched on. Find the number of ways the colours can be programmed so that all the \(3\) colours are used.[3]

In another art installation, \(9\) bulbs, labelled \(A\), \(A\), \(B\), \(B\), \(C\), \(C\), \(D\), \(E\), \(F\), are laid on the floor in a circular arrangement.

4. Find the number of possible arrangements if the 2 \(A\)’s are separated and the 2 \(B\)’s are separated.[3]