Five circles are drawn on a piece of paper and connected as shown. Each circle must be coloured red, blue or green. Two circles connected by a straight line may not be coloured the same. How many different ways are there to colour the circles?
(A) 24
(B) 60
(C) 72
(D) 36
(E) 48

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-094.jpg?height=412&width=406&top_left_y=2185&top_left_x=1293)
We label the five circles as shown in the diagram.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-160.jpg?height=450&width=420&top_left_y=257&top_left_x=907)

We note that there are 3 possible colours and that no two adjacent circles can be coloured the same.

Consider circle $R$. There are three possible colours for this circle.

For each of these colours, there are 2 possible colours for $T$ (either of the two colours that $R$ is not), since it cannot be the same colour as $R$.

Circles $Q$ and $S$ are then either the same colour as each other, or are different colours.

Case 1: $Q$ and $S$ are the same colour

In this case, there are 2 possible colours for $Q$ (either of the colours that $R$ is not) and 1 possibility for $S$ (the same colour as $Q$ ).

For each of these possible colours for $Q / S$, there are two possible colours for $P$ (either of the colours that $Q$ and $S$ are not).

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-160.jpg?height=479&width=631&top_left_y=1414&top_left_x=801)

In this case, there are thus $3 \times 2 \times 2 \times 1 \times 2=24$ possible ways of colouring the circles.

Case 2: $Q$ and $S$ are different colours

In this case, there are 2 possible colours for $Q$ (either of the colours that $R$ is not) and 1 possibility for $S$ (since it must be different from $R$ and different from $Q$ ).

For each of these possible colourings of $Q$ and $S$, there is 1 possible colour for $P$ (since $Q$ and $S$ are different colours, $P$ is different from these, and there are only 3 colours in total).

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-161.jpg?height=472&width=629&top_left_y=160&top_left_x=802)

In this case, there are thus $3 \times 2 \times 2 \times 1 \times 1=12$ possible ways of colouring the circles.

In total, there are thus $24+12=36$ possible ways to colour the circles.

ANSWER: (D)