Question

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.)

Answer 1

Sam can color the sides of the equilateral triangle in 21 different ways.

Step-by-step explanation:

An equilateral triangle has three sides of the same length. In this case, Sam wants to color the three sides of the triangle using five different colors. We need to consider rotations and reflections as the same coloring. To calculate the number of different ways Sam can color the sides, we need to divide the total number of possible colorings by the number of possible rotations/reflections.

First, let's calculate the total number of possible colorings. For the first side, Sam has 5 choices. For the second side, Sam again has 5 choices, regardless of the color chosen for the first side. Similarly, for the third side, Sam has 5 choices regardless of the colors chosen for the first two sides.

So the total number of possible colorings is 5 x 5 x 5 = 125.

Now, let's consider the rotations/reflections. An equilateral triangle has 3 rotational symmetries (rotations by 120 degrees) and 3 reflectional symmetries (flipping the triangle over each side). Each rotation/reflection produces the same coloring. Therefore, to get the number of different colorings, we divide the total number of possible colorings by 6 (3 rotations x 2 reflections).

So the number of different ways Sam can color the sides is 125 / 6 = 20.83 (rounded to the nearest whole number). Therefore, Sam can color the sides of the equilateral triangle in 21 different ways.

Answer 2

Sam can color the three sides of the equilateral triangle in 135 ways

How many ways can Sam color the three sides of the equilateral triangle

To color the three sides of the equilateral triangle, we consider the following cases:

• Coloring with distinct colors
• Coloring with two identical colors

For the distinct colors

Each side of the triangle can be colored independently with one of the five colors.

So, we have

Ways = 5 × 5 × 5

For the identical colors, we have that

Sam can choose two colors for the triangle in ⁵C₂ ways

So, we have

Total = 5 × 5 × 5 + ⁵C₂

Applying the combination formula, we have

Total = 125 + 10

Evaluate

Total = 135

Hence, the total number of ways is 135