<s>[INST] There are 10 sticks of length 1,..,10. How many triangles can be formed There are 10 distinct sticks of length 1,..,10. How many triangles can be formed? I do not know whether there are some counting tricks for this one.
 [/INST] given the triangle inequality and a hint of euclidean geometry, a triangle(possibly degenerate) can have sides $a,b,c$ with $a\leq b\leq c$ if and only if $c< a+b$
There are $\binom{10}{3}$ ways to select three integers between $1$ and $10$
It should be clear there are $k-1$ ways to write the number $k$ as a sum of two different positive numbers
Lets count how many of these selections leave the larger number bigger than the sum of the other two classifying on the larger number $c$
$c=4: (4-1)=3$
$c=5:4+3$
$c=6:5+4+3$
In general the number of selections of number that cannot form a triangle where $c$ is the larger number is $3+4+\dots+(c-1)$ This sum is  $\frac{(c+2)(c-3)}{2}=\frac{c^2-c-6}{2}$
what we would like is $\sum_{4}^{n}\frac{c^2-c-6}{2}=\frac{1}{2}(\sum_4^nc^2-\sum_n^4c-6(n-3))$
We shall simpify this formula using the formulas for the sum of squares and the gaussian sum to get:
$\frac{1}{2}[(\frac{n(n+1)(2n+1)}{6}-14)-(\frac{n(n+1)}{2}-6)-6(n-3)]$
This can be further simplified by the reader.
All you have to do is change $n$ for $10$ and you should hopefully get the number of triangles up to congruence that can be created using those sticks (some will be degenerate though).
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