Forming incongruent triangles

Revision en3, by Samin_Mustakim, 2024-08-29 22:26:35

Problem: There are $$$n$$$ sticks of length $$$1,2,3,...,n$$$. How many incongruent triangles can be formed by using three of the given sticks?

My solution: After solving the problem, I found the number of triangles, $$$x$$$, to be:

$$$\begin{align}x = \sum_{i=1}^{n} i \cdot \left\lceil \frac{i+1}{2} \right\rceil + 2 \cdot \sum_{i=1}^{n} \left\lceil \frac{i+1}{2} \right\rceil - \sum_{i=1}^{n} \left\lceil \frac{i+1}{2} \right\rceil^2 - n(n+1)\newline \end{align}$$$

However, I am not satisfied with this expression. Can it be simplified? (Can we express this equation without using the summation sign?)

Tags combinatorics, counting

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  Rev. Lang. By When Δ Comment
en3 English Samin_Mustakim 2024-08-29 22:26:35 58
en2 English Samin_Mustakim 2024-08-29 22:26:18 91
en1 English Samin_Mustakim 2024-08-29 22:09:35 602 Initial revision (published)