There are 6 Points on xy-Plane.

Question: There are 6 points on xy-plane. Any three points of these 6 don’t lie on the same line. How many unique triangles can be drawn using these 6 points as vertices?

1. 5
2. 10
3. 20
4. 30
5. 60

“There are 6 points on xy-plane. Any three points of these 6 don’t lie on the same line. How many unique triangles can be drawn using these 6 points as vertices?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer

Approach Solution 1

We cans solve this question by drawing the diagram and then counting the triangles. But this method will not give us the accurate answer.

So, we will use the combinations formula to solve this question.

The problem boils down by picking up 3 points out of 6.

So, the answer will be as follows:

\(^6C_3=\frac{6!}{3!3!}=\frac{6*5*4}{3*2}=20\)

Correct option: C

Approach Solution 2

Apart from the combinations method, we can use another method to solve this problem:
Let us organize the information based on 6 unique points: A, B, C, D, E and F. Just maintain alphabetical order and this will make the question solving easy.
AB| ABC, ABD, ABE, ABF (4)
AC| ACD, ACE, ACF (3)
AD| ADE, ADF (2)
AE| AEF (1)

This will give us 10 valid triangles.
Notice the 4-3-2-1 pattern. This sort of countdown will continue.
BC| BCD, BCE, BCF (3)
BD| BDE, BDF (2)
BE| BEF (1)

We can add another 6 valid triangles and move on to point C.
CD| CDE, CDF (2)
CE| CEF (1)
Another 3 valid triangles to add to the tally. All that is left is the final valid combination.
DE| DEF (1)

10 + 6 + 3 + 1 = 20
So, there are 20 valid triangles within the given conditions.

Correct option: C

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