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

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

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