Problem: How many triangles can be formed from a set of $6$ rods of length $1,2,3,4,5,6$ cm taken $3$ at a time?
I managed to solve this problem, by applying both the triangle inequalities and testing for solutions (since there were only $15$ valid cases) and found the answer as $7$. But, is there any way I can develop a more general method to solve this problem for a larger number of rods?
It becomes really difficult to sort through individual cases as the number increases, even to $n=9$; I think my answer then was around $90$ or so. So, any suggestions to make my calculation more efficient? or is there a more elegant way to do this?