Given: “S” has set of points in the plane.
To find: How many number of distinct triangles can be drawn from set S?
Inference: We need three distinct points to form a triangle which is not in a straight line, which means they shouldn’t be collinear.
Analysis of statement 1: The number of distinct points in S is 5.As we think that this statement provides us with the information about how many points are there in set S, to form triangle we need 3 points, therefore the number of triangles formed will be \(= 5C2 = 10.\)
I can say that this statement is a
"trap" where we tend to make mistake by choosing the answer statement 1 is sufficient.
Here we can have two cases:Case1: If all the 5 points are on the same straight line, then we cannot form the triangle.
Case 2: If all the 5 points are not collinear in nature then we can form the triangles.
So we have to be very careful regarding the trap.
Hence statement 1 is not sufficient to answer. We can eliminate options A and D.
Analysis of statement 2: No three of the points in S are collinear.This statement provides us with information about three points in set S are collinear in nature, but we do not have the information about the total number of points. The number triangles are dependent on the total number of points.
Hence statement 2 is not sufficient to answer. We can eliminate option B.
Combining the statements 1 and 2 together; we get:From statement 1: The number of points in set S is “5”
From statement 2: Three points in set S are collinear.
So, we can select 3 points from 5 points to form a triangle.
Number of triangles = \(5C3\)= \(\frac{(5 ×4 ×3)}{(3 ×2 ×1)}\) = 10.
Therefore combining the statements 1 and 2 we get the answer.
So, the correct answer option is “C”.
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