A circle is drawn with 9 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 4 sides is
Answer
Final Answer: The total number of polygons that can be created with at most 4 sides is \(\boxed{210}\).
Step 1 ：A circle is drawn with 9 points on its circumference. A student creates polygons with 3 or more sides by connecting the points with straight lines.
Step 2 ：We need to consider the number of ways to choose points on the circle to form polygons with 3 or 4 sides.
Step 3 ：For a polygon with 3 sides (a triangle), we need to choose 3 points out of 9. This can be done in \(\binom{9}{3}\) ways, which equals to 84.
Step 4 ：For a polygon with 4 sides (a quadrilateral), we need to choose 4 points out of 9. This can be done in \(\binom{9}{4}\) ways, which equals to 126.
Step 5 ：The total number of polygons is the sum of these two quantities, which is \(84 + 126 = 210\).
Step 6 ：Final Answer: The total number of polygons that can be created with at most 4 sides is \(\boxed{210}\).
