Complementary counting solution for mini-question

[professionalbronco]

The total number of ways to color a graph like this
is the same as the number of ways to color a graph like this

minus the number of ways to color a graph like this
.
The number of ways to color a graph shown in graph #2 is just

$$(3\times 2\times 2\times 2)\times (3\times 2\times 2)=288. $$

Remember that there would be a 1/3 probability for these two nodes to have the same color.
So there are a total of \(\dfrac{1}{3}\times 288=96\) "bad" ways.
Thus, we get an answer of

$$(\text{\# of ways to color the original graph})=288-96=\boxed{192}. $$

In other words, this is the same as \(\left(1-\dfrac{1}{3}\right)\times 288=\boxed{192}.\)

[quacker88]

@professionalbronco Really smart use of complementary counting! It's always super cool when a totally different solution is found. Great job!