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How many quadrilateral can be formed from 25 points out of which 7 are collinear
- a)5206
- b)2603
- c)13015
- d)11985
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
How many quadrilateral can be formed from 25 points out of which 7 are...
To solve this problem, we need to understand the concept of quadrilaterals and combinations.
Quadrilaterals are polygons with four sides. In order to form a quadrilateral, we need to select four points out of the given 25 points. However, we are given that 7 points are collinear, which means they lie on the same line.
To calculate the number of quadrilaterals that can be formed, we need to consider different cases based on the collinear points.
Case 1: No collinear points are selected
In this case, we have to select 4 points out of the remaining 25 - 7 = 18 points. The number of ways to select 4 points out of 18 is given by the combination formula: C(18, 4) = 18! / (4! * (18-4)!) = 3060.
Case 2: Exactly 1 collinear point is selected
In this case, we have to select 3 points out of the remaining 25 - 1 = 24 points. The number of ways to select 3 points out of 24 is given by the combination formula: C(24, 3) = 24! / (3! * (24-3)!) = 2024.
Case 3: Exactly 2 collinear points are selected
In this case, we have to select 2 points out of the remaining 25 - 2 = 23 points. The number of ways to select 2 points out of 23 is given by the combination formula: C(23, 2) = 23! / (2! * (23-2)!) = 253.
Case 4: Exactly 3 collinear points are selected
In this case, we have to select 1 point out of the remaining 25 - 3 = 22 points. The number of ways to select 1 point out of 22 is given by the combination formula: C(22, 1) = 22! / (1! * (22-1)!) = 22.
Case 5: Exactly 4 collinear points are selected
In this case, we have to select 0 points out of the remaining 25 - 4 = 21 points. The number of ways to select 0 points out of 21 is given by the combination formula: C(21, 0) = 21! / (0! * (21-0)!) = 1.
Now, to find the total number of quadrilaterals, we sum up the number of quadrilaterals from each case:
Total = Case 1 + Case 2 + Case 3 + Case 4 + Case 5
Total = 3060 + 2024 + 253 + 22 + 1
Total = 5360
Therefore, the correct answer is option 'D' 5360.
Quadrilaterals are polygons with four sides. In order to form a quadrilateral, we need to select four points out of the given 25 points. However, we are given that 7 points are collinear, which means they lie on the same line.
To calculate the number of quadrilaterals that can be formed, we need to consider different cases based on the collinear points.
Case 1: No collinear points are selected
In this case, we have to select 4 points out of the remaining 25 - 7 = 18 points. The number of ways to select 4 points out of 18 is given by the combination formula: C(18, 4) = 18! / (4! * (18-4)!) = 3060.
Case 2: Exactly 1 collinear point is selected
In this case, we have to select 3 points out of the remaining 25 - 1 = 24 points. The number of ways to select 3 points out of 24 is given by the combination formula: C(24, 3) = 24! / (3! * (24-3)!) = 2024.
Case 3: Exactly 2 collinear points are selected
In this case, we have to select 2 points out of the remaining 25 - 2 = 23 points. The number of ways to select 2 points out of 23 is given by the combination formula: C(23, 2) = 23! / (2! * (23-2)!) = 253.
Case 4: Exactly 3 collinear points are selected
In this case, we have to select 1 point out of the remaining 25 - 3 = 22 points. The number of ways to select 1 point out of 22 is given by the combination formula: C(22, 1) = 22! / (1! * (22-1)!) = 22.
Case 5: Exactly 4 collinear points are selected
In this case, we have to select 0 points out of the remaining 25 - 4 = 21 points. The number of ways to select 0 points out of 21 is given by the combination formula: C(21, 0) = 21! / (0! * (21-0)!) = 1.
Now, to find the total number of quadrilaterals, we sum up the number of quadrilaterals from each case:
Total = Case 1 + Case 2 + Case 3 + Case 4 + Case 5
Total = 3060 + 2024 + 253 + 22 + 1
Total = 5360
Therefore, the correct answer is option 'D' 5360.
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Community Answer
How many quadrilateral can be formed from 25 points out of which 7 are...
otal number of quadrilateral combination possible if none of the points are collinear = 25C4 = 12650
If we form a geometry by joining any three points out of seven collinear points and one point from 18 non collinear points, it will give us a triangle instead of quadrilateral. So we have to eliminate number of combinations which can be formed in this way, which is 7C3 x 18C1 = 35 x 18 = 630
We also can't form quadrilateral if we choose all four vertices of quadrilateral to be any 4 points from 7 collinear points. It will come out to be a straight line. So we have to eliminate such combinations also. Which is 7C4 = 35
So net number of possible quadrilaterals = 12650 - 630 - 35 = 11985
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