Triangle-Ception, Don't Lose Your Count!

Probability Level pending

Calvin was given 2016 line segments with lengths of $1, 2, 3, 4,\ldots , 2015, 2016$ respectively and he had to pick three random lines to form a triangle. Determine the possible number of triangles Calvin could form.

Details and assumptions

$\bullet$ All the lengths are in the same system of units.

This problem is inspired by a much simpler problem taken from a middle school textbook.

The answer is 681271752.

1 solution

Kunal Verma
Dec 28, 2015

Okay this was something very tedious for me and I'm expecting a better solution.

I opted for total cases - illegal cases.

Total cases= $\left(\begin{array}{c}2016\\ 3\end{array}\right)$

Illegal cases :-

For smallest side= $1$ :- $2014 \ + \ 2013 \ + \ ... \ + \ 1$ = $S( \ 2014)$

For smallest side = $2$ :- $2012 \ + \ 2011 \ + \ ... \ + \ 1$ = $S( \ 2012)$

And so on till:-

For smallest side = $1007$ :- $2 \ + \ 1$ = $S( \ 2)$

Thus total number of illegal cases = $S( \ 2) \ + \ S( \ 4) \ + \ ... \ S( \ 2014) \ = \sum_{x=1}^{1007}(1008 \ - \ x) \ (\ 4x \ - \ 1\ )$

This easily computes to $682286808$

Hence, total number of legal triangles = $\left(\begin{array}{c}2016\\ 3\end{array}\right)\ - \ 682286808 \ = \boxed{681271752}$

@Leah Smith Can you post your solution?

Kunal Verma - 5 years, 5 months ago

It's not that tedious . I did it same way and I think it's easiest one. However, you can always come with better solution.

Pranjal Prashant - 5 years, 5 months ago

@Kunal Verma @Pranjal Prashant Not that tedious but not worth only 10 pts either. I mean they also have 100 pts for pending problems right? I myself really appreciate you two's effort for actually doing this problem knowing it's only worth 10 pts!

Leah Smith - 5 years, 5 months ago

10 points are given for unseeded problems. seeded problems have 100 pts. Some of my questions in physics took 3 months to get rated and now they are all in level 4 or 5. We will get the points as soon as this problem gets rated. :)

Pranjal Prashant - 5 years, 5 months ago

Sorry for the late reply. Well, this happened to me once before because I had myself marked my problem as unrated while posting it. And it got stuck up with 10 pts forever. But since you've posted questions before, it's highly unlikely you would've done that. Maybe you can ask someone from the staff about it. And points don't really matter if a problem is good, do they? c:

Kunal Verma - 5 years, 5 months ago

@Kunal Verma I will sort it out and post later. But I don't get it. The problem is clearly stated. There has been solvers, solutions to confirm the answer. Yet it's still only given 10 points, which make people just look at it, not even attemp to do such a tedious task...

Leah Smith - 5 years, 5 months ago

It's level is undecided yet. It will change after a few more people attempt it and depending on the number of solvers, it will be assigned a level. You can see that stated at the top.

Kunal Verma - 5 years, 5 months ago

It will get rated as soon as more people will attempt it. This one will probably have rating 250 or even higher.

Pranjal Prashant - 5 years, 5 months ago

Triangle-Ception, Don't Lose Your Count!

This problem is inspired by a much simpler problem taken from a middle school textbook.

The answer is 681271752.

1 solution

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