S is a set of points in the plane. How many distinct triangles can be

l got A. What does collinear mean?

Collinear points are those that lie on the same straight line. BTW the correct answer is C, not A. Check the solutions above and ask if anything remains unclear.

Does distinct pint not mean that points are differently located.?
I marked the A assume the above.
please make it clear.

From (1) the points are distinct but 3 or more from them can be on the same line (collinear), thus they won't form a triangle.

I marked E... considering how can we be sure of the distances between the points would make valid Triangles?? What am I missing??

ANY 3 points on a plane that are not collinear form a triangle.

Example: if the distances between the 3 points are 5,6 & 13? Here they won't form a triangle right? Since 5+6 < 13?

You cannot have this case. If the distance from A to B is 5 and the distance from B to C is 6, then the distance C to A cannot be more than 11.

We are given that S is a set of points in the plane and we must determine how many distinct triangles can be drawn with three of the points in S as vertices. So essentially, we must determine how many distinct triangles can be drawn with the points provided.

Statement One Alone:

The number of distinct points in S is 5.

Using the information in statement one, it may be tempting to conclude that the number of triangles that can be drawn is 5C3 = (5 x 4 x 3)/3! = 10 triangles. However, because we do not know the positioning of the points, we cannot actually say that 10 distinct triangles can be created. Let’s say, for instance, that all the points were collinear, which means that they are all located on one line. If that were the case, we would not be able to create any triangles. Thus, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Note: We were able to determine that 10 triangles could be formed with 5 points if and only if no 3 points are collinear. Only then would the number of triangles be 5C3 = 10.

Statement Two Alone:

No three of the points in S are collinear.

Using the information in statement two, we cannot answer the question because we do not know how many points are in S. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two we know that we have 5 points in the plane and that no three points are collinear. Thus, we can determine that the number of triangles that can be created in the plane is 5C3 = 10.

Answer: C

I know that this is over a year late but I just want to clarify that if one were to connect 3 points (ABC) with the distances of 5(AB), 6(BC), and 13(AC), the only possible way is to make all 3 points collinear. Which would thus not form a triangle.

Responding to a pm:

S is a set of points. To make a triangle, we need 3 distinct points such that the 3 do not lie in a straight line (i.e. are not collinear). When you join 3 points which are in a straight line, you get a line, not a triangle.

(1) The number of distinct points in S is 5.

We know that we have 5 points. but what if all 5 are in a straight line? We won't be able to make any triangles.
If they are not in a straight line, we will be able to make triangles. Hence, this statement alone is not sufficient.

(2) No three of the points in S are collinear.

We know that the points are not collinear but how many points do we have? The more the number of points, the more the number of triangles.

Using both statements, we know that we have 5 points, no 3 of which are collinear. So we can select any 3 points out of 5 and make a triangle out of them. No of triangles we can make = 5C3 = 10 triangles.

Can you please write formula to calculate the number of triangles if only 3 points are coliner

Hello

Ok, here's a general formula for calculating possible number of triangles.

If there are N points in a plane, and no three of them are collinear, then the number of possible triangles = N(C)3. (number of possible selections of 3 objects from N objects)

If there are N points in a plane, and X of them are collinear, then the number of possible triangles = N(C)3 - X(C)3. (subtracting number of possible selections of 3 objects from X FROM number of possible seelctions of 3 objects from N).

Now to answer your question: IF we were given that there are 5 points in a plane, out of which 3 are collinear, then number of possible triangles would have been = 5C3 - 3C3 = 10 - 1 = 9. Thus 9 triangles would have been possible.

Hi Scott, I understand the logic behind the answer and got C as my answer. However, why does 5C3 work? I don´t understand that concept and maybe it will come in the exame in a PS type.
Thanks!

If we know that no three of the points are on the same line, then the number of distinct triangles that can be drawn using the five points is 5C3. This is because if no three of the points are on the same line, then any three points we choose from among the five points will form a triangle, and the number of ways we can choose those three points is given by 5C3.

It might actually be helpful to consider the question "when will the three points we choose won't form a triangle?". If we have three points on a plane, the only way for those points to not form a triangle is if they are collinear (meaning there is a line containing all the three points). As we are told by statement two that that is not the case, we know for sure that the number of triangles that can be formed using those five points is 5C3.

Answer: Option C

Video solution by GMATinsight

I know the answer is C by now, but annoyed that the definition of "distinct" triangle is not clear, as without knowing the actual coordination of the points, you will not know whether you will form same triangle. Hence a contending ans E