Graph Theory - Application of Kirchoff's Matrix Tree Theorem Calculate the number of spanning trees of the graph that you obtain by removing one edge from $K_n$. 
(Hint: How many of the spanning trees of $K_n$ contain the edge?)
I know the number is $(n-2)n^{n-3}$ and that Kirchoff's matrix tree theorem applies but how do I show this?

All edges of $K_n$ are identical. So if there are $n^{n-2}$ spanning trees of $K_n$, and each includes $n-1$ edges out of $\binom n2$, then each edge is included in $$\frac{n-1}{\binom n2} \cdot n^{n-2} = \frac2n \cdot n^{n-2} = 2n^{n-3}$$ spanning trees.
(Normally, this would be "included in an average of $2n^{n-3}$ spanning trees", but because all edges of $K_n$ are identical, all of them are contained in exactly the average number.)
