How many different triangles are there in $K_5$?
The Answer is 35.(The Moscow Mathematics Puzzle)
Then I asked what about $K_6$, $K_7$ and so on ...?
With my intuition I arrived at this conjecture
Conjecture: The number of different triangles in a complete graph of order $n$ ($K_n$) is
\begin{cases} \frac{n^2(n-2)^2(n-1)^2)}{6^2},&\mbox{ where } \binom{n}{3} \mbox{ is odd.}\\ \frac{(n^2-n)(n^2-4)(n^2-5n+12)}{12^2} ,&\mbox{where }\binom{n}{3} \mbox{is even.} \end{cases} What do you say about this?