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

• Hold up-

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

• @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.