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From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
SolutionsSolution 1We will use constructive counting to solve this. There are cases: Either all points are adjacent, or exactly points are adjacent. If all points are adjacent, then we have choices. If we have exactly adjacent points, then we will have places to put the adjacent points and also places to put the remaining point, so we have choices. The total amount of choices is .Thus our answer is Solution 2We can decide adjacent points with choices. The remaining point will have choices. However, we have counted the case with adjacent points twice, so we need to subtract this case once. The case with the adjacent points has arrangements, so our answer is Solution 3 (Stars and Bars)Let point of the triangle be fixed at the top. Then, there are ways to chose the other 2 points. There must be spaces in the points and points themselves. This leaves 2 extra points to be placed anywhere. By stars and bars, there are 3 triangle points (n) and extra points (k-1) distributed so by the stars and bars formula, , there are ways to arrange the bars and stars. Thus, the probability is .Video Solutionshttps://youtu.be/5UojVH4Cqqs?t=2678 https://www.youtube.com/watch?v=VNflxl7VpL0 https://youtu.be/YeYDixFXsvA ~savannahsolver gg See AlsoThe problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
There are a few conflicting answers here. The correct count is 16. One approach (in a comment by @juantheron) is to consider all triples of vertices and subtract those that share exactly one side with the octagon and those that share exactly two sides with the octagon: $$\binom{8}{3} - 8\cdot 4 - 8 = 56 - 32 - 8 = 16.$$ A similar approach is to use the inclusion-exclusion principle. Start with all triples of vertices, subtract (an overcount of) those that share at least one side, and add back in those that share two sides: $$\binom{8}{3} - 8\cdot 6 + 8 = 56 - 48 + 8 = 16.$$ Here is the explicit list of 16: ACE, ACF, ACG, ADF, ADG, AEG, BDF, BDG, BDH, BEG, BEH, BFH, CEG, CEH, CFH, DFH Anyone who claims a count higher than 16, please show me one that is not on this list. |