Final answer:
To find the number of distinct polygons that can be formed with seven non-collinear points, we sum the combinations of choosing 3 to 7 points from the seven. This involves the use of combinatorics to calculate the possible ways to form triangles up to heptagons.
Step-by-step explanation:
If we have seven distinct points on a plane with no three points being collinear, we can determine the number of different polygons that can be formed by these points. A polygon is defined as a closed plane figure with at least three straight sides and angles, typically with a fixed number of sides. Starting from the simplest polygon, a triangle, we can form a polygon with any subset of three or more points.
To calculate the number of different polygons, we can use the concept of combinations from combinatorics. We use the combination formula ∂(n, r) = n! / (r!(n - r)!), where n is the total number of points and r is the number of points chosen to form a polygon. For our seven-point problem, we consider polygons with 3 up to 7 vertices. Therefore, the number of different polygons can be found by summing the combinations of choosing 3, 4, 5, 6, and 7 points from the seven.
Calculating the number of polygons:
- Triangles: ∂(7, 3)
- Quadrilaterals: ∂(7, 4)
- Pentagons: ∂(7, 5)
- Hexagons: ∂(7, 6)
- Heptagons: ∂(7, 7)
By summing these values, we can arrive at the total number of distinct polygons that can be formed.