Question: You are given n sticks of lengths 1 to n unit respectively. Using three different sticks how many different triangles can you form?
I have found a formula on the internet. That is if n is odd triangle=(n-1)*(n-3)*(2n-1)/24 else triangle=n*(n-2)*(2n-5)/24
But I could not prove this or find any proof. Can anyone provide with a proof of this formula? How to derive this? Or share if there are other ways to solve this problem? Thanks in advance.
It can be proved by induction.
; if n0 is even then the number is 
; in both cases the first term is the number of triangles with lengthes ≤ n0 and the second term is the number of triangles with one length that equal to n. By simplifying expressions, you can get needed formula
For n = 4 the number is 1. For n = n0 + 1, if n0 is odd then the number is
So we're assuming degenerate triangles are allowed? Otherwise I think the only valid triples is {2, 3, 4}. In fact, the given formula n(n - 2)(2n - 5) / 24 gives 1 as well.
oh i made mistake, edited now