??? That just completely ignores the possibility of the two lower nodes being different colors, which isn't right, no?

[The Blade Dancer]

Hold up-

That just completely ignores the possibility of the two lower nodes being different colors, which isn't right, no?

[debbie]

@The-Blade-Dancer Hi again! The solution is counting these two separate cases: Case 1.) where the lower two second-from-bottom nodes are the $\textcolor{red}{\text{same}}$ color, and Case 2.) where the lower two second-from-bottom nodes are $\textcolor{red}{\text{different}}$ color.

For Case 1: There were $48$ ways to color the top six nodes (with the two bottom nodes the same color)

For Case 2: There were $96$ ways to color the top six nodes (with the two bottom nodes different color).

In the final formula, the $48$ is multiplied by $2$ because there are two options for the very bottom node color (e.g. $\textcolor{green}{\text{green}}$ and $\textcolor{orange}{\text{yellow}.})$ The $96$ isn't multiplied by anything, because if you already have two colors for the second-from-bottom nodes, then there's only one choice for the very last bead's color (in this example, $\textcolor{orange}{\text{yellow}}).$

 

 
 
 
$$48 \times 2 + 96 = \boxed{192}$$

That's how the answer was found.