+287 Sam wants to color the three sides of an equilateral triangle. He has five different colors to choose from. In how many different ways can Sam color the sides of the triangle? (Two colorings are considered the same if one coloring can be rotated and/or reflected to obtain the other coloring.)
+506 At first glance, it might seem like there are 5 options for each side and so a total of 5 * 5 * 5 = 125 ways to color the triangle. However, this overcounts the number of colorings because rotating or reflecting the triangle can create what seems like a different coloring.
We can consider two cases:
Case 1: All three sides are the same color.
There are 5 choices Sam can make for this single color (since all three sides are the same).
Case 2: Two sides are the same color and the third side is a different color.
Here, we need to consider how many ways Sam can choose the two colors and how many ways he can arrange them.
Choosing the colors: There are 5 choices for the color used twice and 4 remaining choices for the different color (since one color is already chosen for the two sides). So, there are 5 * 4 = 20 ways to choose the colors.
Arranging the colors: It might seem like there are 2 ways to arrange the colors (one color for two sides and another for the remaining side). However, rotating the triangle flips the arrangement. So, if we consider both rotations as the same coloring, then there is only 1 way to arrange the colors.
Therefore, for Case 2, the total number of colorings is 20 (choosing colors) * 1 (arranging colors) = 20.
Total Colorings:
Adding the number of colorings for both cases:
Total Colorings = Case 1 + Case 2 = 5 (all same color) + 20 (two same, one different) = 25
So, Sam can color the triangle in 25 different ways.
