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Source: 2021 Fall AMC 10B #23

November 17, 2021
AMC 10AMCAMC 10 B

Problem Statement

Each of the 55{ } sides and the 55{ } diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
(<spanclass=latexbold>A</span>)23(<spanclass=latexbold>B</span>)105128(<spanclass=latexbold>C</span>)125128(<spanclass=latexbold>D</span>)253256(<spanclass=latexbold>E</span>)1(<span class='latex-bold'>A</span>)\: \frac23\qquad(<span class='latex-bold'>B</span>) \: \frac{105}{128}\qquad(<span class='latex-bold'>C</span>) \: \frac{125}{128}\qquad(<span class='latex-bold'>D</span>) \: \frac{253}{256}\qquad(<span class='latex-bold'>E</span>) \: 1