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There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:
- a)120
- b)124
- c)116
- d)112
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
There are 10 points in a plane, no three of which are in the same str...
To find the total number of triangles that can be formed with the given set of points, we can break down the problem into three cases:
Case 1: Triangles formed by selecting three non-collinear points
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Case 3: Triangles formed by selecting three points from the collinear set
Now let's analyze each case in detail:
Case 1: Triangles formed by selecting three non-collinear points
Since there are 10 points in the plane and no three points are collinear, we can choose any 3 points out of the 10. The number of ways to choose 3 points out of 10 is given by the combination formula C(10, 3) = 10! / (3! * (10-3)!) = 120.
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Since there are 4 points that are collinear, we can choose any 2 points out of the remaining 6 non-collinear points and any 1 point from the collinear set. The number of ways to choose 2 points out of 6 is given by the combination formula C(6, 2) = 6! / (2! * (6-2)!) = 15. And since we have 4 points in the collinear set, we have 4 choices for the third point. Therefore, the total number of triangles in this case is 15 * 4 = 60.
Case 3: Triangles formed by selecting three points from the collinear set
Since there are 4 points that are collinear, we can choose any 3 points from this set. The number of ways to choose 3 points out of 4 is given by the combination formula C(4, 3) = 4! / (3! * (4-3)!) = 4.
Therefore, the total number of triangles that can be formed is the sum of the triangles in each case: 120 + 60 + 4 = 184.
However, we need to subtract the triangles formed by selecting all 3 points from the collinear set, as these triangles are not valid according to the given conditions. There is only 1 such triangle. Therefore, the final answer is 184 - 1 = 183.
Hence, the correct option is (C) 116.
Case 1: Triangles formed by selecting three non-collinear points
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Case 3: Triangles formed by selecting three points from the collinear set
Now let's analyze each case in detail:
Case 1: Triangles formed by selecting three non-collinear points
Since there are 10 points in the plane and no three points are collinear, we can choose any 3 points out of the 10. The number of ways to choose 3 points out of 10 is given by the combination formula C(10, 3) = 10! / (3! * (10-3)!) = 120.
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Since there are 4 points that are collinear, we can choose any 2 points out of the remaining 6 non-collinear points and any 1 point from the collinear set. The number of ways to choose 2 points out of 6 is given by the combination formula C(6, 2) = 6! / (2! * (6-2)!) = 15. And since we have 4 points in the collinear set, we have 4 choices for the third point. Therefore, the total number of triangles in this case is 15 * 4 = 60.
Case 3: Triangles formed by selecting three points from the collinear set
Since there are 4 points that are collinear, we can choose any 3 points from this set. The number of ways to choose 3 points out of 4 is given by the combination formula C(4, 3) = 4! / (3! * (4-3)!) = 4.
Therefore, the total number of triangles that can be formed is the sum of the triangles in each case: 120 + 60 + 4 = 184.
However, we need to subtract the triangles formed by selecting all 3 points from the collinear set, as these triangles are not valid according to the given conditions. There is only 1 such triangle. Therefore, the final answer is 184 - 1 = 183.
Hence, the correct option is (C) 116.
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Community Answer
There are 10 points in a plane, no three of which are in the same str...
Number of triangles formed joining the 10 points taken 3 at a time = 10C3 = 120
Number of triangles formed joining the 4 points taken 3 at a time = 4C3 = 4
But four collinear points cannot form a triangle when taken 3 at a time
So total number of required triangle = 120 - 4 = 116.
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There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer?
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