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- In a plane, there are 20 points. Five points lie on a straight line, and the remaining fifteen points do not lie on the same straight line. Determine the total number of unique quadrilaterals that
In a plane, there are 20 points. Five points lie on a straight line, and the remaining fifteen points do not lie on the same straight line. Determine the total number of unique quadrilaterals that can be formed by selecting four points from these 20 points, such that no three points are collinear.
4840
5845
1275
3600
Answers (1)
Case 1: Selecting all four points from the five points that lie on a straight line
In this case, we can select four points from the five points on the line in C(5, 4) = 5 ways. However, since three points on a line are collinear, we cannot form a valid quadrilateral in this case.
Case 2: Selecting three points from the five points on the line and one point from the remaining fifteen points
In this case, we can select three points from the five points on the line in C(5, 3) = 10 ways.
We can select one point from the remaining fifteen points in C(15, 1) = 15 ways.
Total options for Case 2 = 10 15 = 150
Case 3: Selecting two points from the five points on the line and two points from the remaining fifteen points
In this case, we can select two points from the five points on the line in C(5, 2) = 10 ways.
We can select two points from the remaining fifteen points in C(15, 2) = 105 ways.
Total options for Case 3 = 10 105 = 1050
Case 4: Selecting one point from the five points on the line and three points from the remaining fifteen points
In this case, we can select one point from the five points on the line in C(5, 1) = 5 ways.
We can select three points from the remaining fifteen points in C(15, 3) = 455 ways.
Total options for Case 4 = 5 455 = 2275
Case 5: Selecting four points from the remaining fifteen points that do not lie on the line
In this case, we can select four points from the remaining fifteen points in C(15, 4) = 1365 ways.
Total options for Case 5 = 1365
Total number of unique quadrilaterals = Total options for Case 2 + Total options for Case 3 + Total options for Case 4 + Total options for Case 5 = 150 + 1050 + 2275 + 1365 = 4840
Therefore, the total number of unique quadrilaterals that can be formed by selecting four points from the given 20 points, such that no three points are collinear, is 4840.
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