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A circle is drawn with 7 points on its circumference, as shown in the diagram below. A student creates polygons with 3 or more sides by connecting the points with straight lines. The number of polygons that can be created that have at most 5 sides

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a) where is the diagram?

b) the student has 7 points to work with but in this case can choose only 5. Assuming that the points are evenly spaced (this is where the diagram would be useful), the number of 5-sided polygons he can create is 7C5, (7 choose 5), which is equal to 7!/5!(2!) = 21. He can create 21 different pentagons
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Assume that for any set of 5 points, there is only 1 polygon that can be drawn. That is, the polygons are “simple” and the sides don’t intersect. Then, selecting five points is the same as selecting a polygon. But, that’s the same as selecting the two points that aren’t in the polygon. To pick those points, pick one at random (7 ways to do this) and then pick another (6 ways to do this). That gives 7x6=42 ways to select pairs of point. But, selecting A then B is the same as selecting B then A, so we are double counting by a factor of 2, and so the final answer is 42/2 = 21.

Alternatively, use the formula 7C5 = 7!/(5!2!) = 21 to go directly to the answer.

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