Sam wants to color the three sides of an equilateral triangle. He has two 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.)
Sam can color the sides of the triangle in two different ways.
To solve this problem, let's break it down step by step:
Step 1: Understand the problem.
The problem states that Sam wants to color the three sides of an equilateral triangle using two different colors. We need to find out the number of different ways Sam can color the sides, considering that two colorings are considered the same if one coloring can be rotated and/or reflected to obtain the other coloring.
Step 2: Identify the key concepts.
The key concept here is that the equilateral triangle can be rotated or reflected to obtain the same coloring. This means that we need to take into account all the different possible orientations of the triangle.
Step 3: Analyze the problem.
To analyze the problem, we need to consider the different cases where the coloring can be the same despite the rotation or reflection of the triangle. In other words, we need to find out how many distinct colorings can be obtained by rotation or reflection.
Step 4: Solve the problem.
To solve the problem, let's consider the following cases:
Case 1: All three sides have the same color.
In this case, there is only one distinct coloring because rotating or reflecting the triangle will not change the coloring.
Case 2: Two sides have the same color and one side has a different color.
In this case, there are two distinct colorings because we have two possible ways to choose which two sides will have the same color.
Case 3: All three sides have different colors.
In this case, there are also two distinct colorings. To understand why, imagine coloring one side of the triangle with color A. The other two sides can be colored with color B and color C in two different ways: (B,C) or (C,B). Again, rotating or reflecting the triangle does not change the coloring.
Step 5: Calculate the total number of distinct colorings.
To calculate the total number of distinct colorings, we sum up the number of distinct colorings from each case:
Total number of distinct colorings = Number of colorings from Case 1 + Number of colorings from Case 2 + Number of colorings from Case 3
= 1 + 2 + 2
= 5
Therefore, there are 5 different ways Sam can color the sides of the equilateral triangle.
To solve this problem, we can consider the different cases:
Case 1: All three sides are the same color.
In this case, Sam can choose one of the two available colors for the sides. So there are 2 possible colorings.
Case 2: Two sides are the same color, and the third side is a different color.
In this case, Sam can choose 1 color for the two sides that are the same, and the other color for the remaining side. The color of the remaining side can be chosen in two ways. So there are 2*2 = 4 possible colorings.
Case 3: All three sides have different colors.
In this case, Sam can choose 1 color for each side, and the order doesn't matter. So the number of possible colorings is the number of ways to choose 3 colors out of the 2 available colors, without considering the order. This can be calculated using combinations, denoted as C(n, r), where n is the total number of colors and r is the number of colors to be chosen.
C(2, 3) = 2! / (3!(2-3)!) = 2! / (3!(-1)!) = 2! / 3! = (2*1) / (3*2*1) = 2/6 = 1/3
So there is only 1 possible coloring in this case.
Therefore, the total number of different ways Sam can color the sides of the equilateral triangle is 2 + 4 + 1 = 7.