Problem Solving

ANMOL UPADHYAY wrote:how many triangles can be obtained by joining 12 points
5 of which are collinear ?

Let's pretend for a moment that there are no collinear points (i.e., no points on the same line).
In this case, we can create a triangle by selecting any 3 points as vertices. So, the question now becomes, "In how many different ways can we select 3 points?"

Since the order of the selected points does not matter, we can answer this question using combinations.
We can select 3 points from 12 points in 12C3 ways (= 220 ways).

Aside: If anyone is interested, we have a free video on calculating combinations (like 12C3) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

From here, we need to recognize that some of our 3-point selections do not create triangles. If those 3 points are collinear, we won't get a triangle using the 3 points as vertices.

So, how many of those 3-point selections do not create triangles?
Well, in how many ways can we select 3 points from the 5 collinear points?
Once again, the order of the selected points does not matter, so we can use combinations.
We can select 3 points from the 5 collinear points in 5C3 ways (= 10 ways).

So, the total number if different triangles possible = 220 - 10 = 210

Cheers,
Brent