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# Counting

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I don't get this

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.)

Feb 18, 2023

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To determine the number of different ways Sam can color the sides of an equilateral triangle using two different colors, we can consider the following cases:

Case 1: All three sides have the same color. In this case, there are two options: either all sides are colored with the first color, or all sides are colored with the second color. So, there are 2 ways to color the triangle in this manner.

Case 2: Two sides have the same color, and the third side is a different color. We need to consider the permutations of colors within this configuration. There are two options for the color of the two sides that are the same, and two options for the color of the third side. So, there are 2 x 2 = 4 ways to color the triangle in this manner.

Case 3: All three sides have different colors. Since Sam has two different colors to choose from, there are two options for each side. So, there are 2 x 2 x 2 = 8 ways to color the triangle in this case.

Therefore, the total number of different ways Sam can color the sides of the equilateral triangle, taking into account the symmetries, is 2 + 4 + 8 = 14.

Jun 28, 2023