IF THERE ARE 7 DISTINC POINTS ON THE PLANE WITH NO THREE OF WHICH ARE COLLINEAR, HOW MANY DIFFERENT POLYGONS CAN BE POSSIBLY FORMED?
Polygons with 3 vertices, (triangle) = C(7,3) = 35
polygons with 4 vertices = C(7,4) = 35
polygons with 5 vertices = C(7,5) = 21
polygones with 6 vertices = C(7,6) = 7
polygons with 7 vertices = 1
add them up
To determine the number of different polygons that can be formed using 7 distinct points on a plane, we can follow these steps:
1. Determine the number of ways to choose the vertices of the polygon: Since no three points are collinear, any polygon formed will have at least 3 vertices. We can choose the vertices in C(n, r) ways, where n is the total number of points (7 in this case) and r is the number of vertices we want to choose. Since we want to form polygons, we will choose 3, 4, 5, 6, or 7 vertices.
2. Calculate the number of polygons for each possible number of vertices:
- For choosing 3 vertices, there is only 1 way to form a triangle with the given points.
- For choosing 4 vertices, there are C(7, 4) ways to form a quadrilateral.
- For choosing 5 vertices, there are C(7, 5) ways to form a pentagon.
- For choosing 6 vertices, there are C(7, 6) ways to form a hexagon.
- For choosing 7 vertices, there is only 1 way to form a heptagon with the given points.
3. Sum up the number of polygons: Add up the number of polygons for each possible number of vertices.
- 1 triangle + C(7, 4) quadrilaterals + C(7, 5) pentagons + C(7, 6) hexagons + 1 heptagon = 1 + 35 + 21 + 7 + 1 = 65.
Therefore, there are a total of 65 different polygons that can be formed using 7 distinct points on the plane, with no three of which are collinear.
To find the number of different polygons that can be possibly formed with 7 distinct points on a plane, where no three points are collinear, we can use the concept of combinations.
A polygon can be formed by connecting the points in different orders. We need to consider the number of sides of the polygon, which can range from a minimum of 3 to a maximum of 7 (since we have 7 points).
To form a polygon with n sides, we need to choose n points out of the 7 given points. The number of ways to choose n points out of 7 is given by the combination formula:
C(n, r) = n! / ((n - r)! * r!)
Where n! denotes the factorial of n.
Now, let's calculate the number of polygons for each possible number of sides (n):
For n = 3:
C(7, 3) = 7! / ((7 - 3)! * 3!) = 7! / (4! * 3!) = 7 * 6 * 5 / (3 * 2 * 1) = 35
For n = 4:
C(7, 4) = 7! / ((7 - 4)! * 4!) = 7! / (3! * 4!) = 7 * 6 * 5 * 4 / (4 * 3 * 2 * 1) = 35
For n = 5:
C(7, 5) = 7! / ((7 - 5)! * 5!) = 7! / (2! * 5!) = 7 * 6 * 5 / (5 * 4 * 3 * 2 * 1) = 21
For n = 6:
C(7, 6) = 7! / ((7 - 6)! * 6!) = 7! / (1! * 6!) = 7 * 6 / (6 * 5 * 4 * 3 * 2 * 1) = 7
For n = 7:
C(7, 7) = 7! / ((7 - 7)! * 7!) = 7! / (0! * 7!) = 1
Now, summing up all the possibilities for different numbers of sides:
35 + 35 + 21 + 7 + 1 = 99
Therefore, there are a total of 99 different polygons that can be possibly formed with 7 distinct points on the plane, where no three of them are collinear.