Okay so let me define some terms before I ask my problem:
Let $K_n$ denote the complete graph on $n$ vertices with $n\geq 2$ and let $C_3$ be a cycle of length $3$ (a triangle).
Suppose $x,y,z$ are distinct vertices in $K_n$:
How many copies of $C_3$ are there in $K_n$ which contain at least one of the edges $xy$, $yz$ and $xz$?
My thoughts are like this:
Picking any edge from $xy$, $yz$ and $xz$ gives $n-2$ possible copies of $C_3$ (Just select the two other edges to meet at the same point and since this is a complete graph there must be $n-2$ of these.)
This is the same for the other two edges that you did not select.
So in total there are $3(n-2)$ copies but this overcounts so we need to remove the overcounting.
Choosing two edges from $xy$, $yz$ and $xz$ determines the triangle $xyz$ and so we need to remove this $3$ times and then add it back one last time. (See inclusion exclusion principle for $3$ sets), thus there are $3n-8$ total triangles in $K_n$.
Is this along the right lines?