Question

In how many ways can we color the $8$ vertices of an octagon each red, green, or blue, so that no two adjacent vertices are the same color

Answer 1

To color the vertices of an octagon with three different colors, such that no two adjacent vertices are the same color, there are 48 different ways.

Step-by-step explanation:

To color the vertices of an octagon with three different colors, such that no two adjacent vertices are the same color, we can use a tree diagram to enumerate all the possible color combinations.

Starting with the first vertex, there are three choices for the color. For the second vertex, there are two remaining choices, since we cannot use the same color as the first vertex. For the third vertex, there are again two choices, and so on.

Using this method, we can find that there are a total of 48 different ways to color the vertices of the octagon.