2019 Mock AMC 10B Problems/Problem 14
Solution 1
There are five points, and to make a triangle we need 3, so the number of ways is just 
This is option choice (A).
- Armpit Splash 💦
- Edits by Continuous_Pi
Note: This solution is wrong because we must consider when a subset of the points are collinear, creating degenerate triangles.
Note 2: It doesn't matter the problem is just bad and should say that no three of the points are collinear or something because if there were degenerate triangles the answer would be less than 10, but that's not an answer choice so it doesnt matter - lucaswujc
Note 3: A degenerate triangle is just a line, and we do not call lines triangles very often. And if we neglected degenerate triangles, the answer would not be less than 10. It will be greater, due to more cases being added.
Solution 2 (assuming degenerate triangles are not counted as actual triangles in this problem)
Let's split the problem into 4 cases: <3 points are collinear, 3 points (and no more than that) at a time are collinear, 4 points (and no more than that) at a time are collinear, and 5 points at a time are collinear
Case 1: <3
That will just be 5C3, as degenerate triangles cannot form here. So, it is just 10 for this case
Case 2: 3 at a time (and no more) are collinear
There are 2 ways this can go. One way is that there is only 1 line going through 3 points. The other way is that there are 2 lines going through 3 points. That is made possible through the 2 lines intersecting at one points. Anything more is impossible. So, for the case where there is only 1 line, that will be 5C3 - 1, the 1 being the combination of 3 that is collinear. Apply similar logic to the other case, you get 5C3 - 2. In total, we arrive at 8+9 = 17. So for this case, it is 17.
Case 3: 4 at a time (and no more) are collinear
There can only be 1 line going through 4 points at a time. So, that will be 5C3 - 4C3, the 4C3 being the amount of ways you can choose 3 from that set of 4 collinear points. So we arrive at 10-4 = 6.
Case 4: 5 at a time are collinear
Here, every single combination of 3 points chosen will be collinear and will result in a degenerate triangle. So, here, we have 0
In total, the we have 10+17+6 = 33. Hence the answer is E.33 - Solution by cowcheese
