Let $V=(1,2,...,n), n>5$ be a set of vertices. How many tagged trees are there on V such that the vertices 1,2,3 are leaves and the minimum path length between every 2 of them is greater than 2?
What I've tried: I said that without 1,2 and 3 there are $(n-3)^{n-5}$ tagged trees on V without 1,2,3. So now we need to connect 1,2,3 to the tree with one edge each (so it will be a leaf) and it can't be the same vertex in the tree every time because this will result a length of 2.
So this will give: $(n-3)^{n-5}\cdot(n-3)\cdot(n-4)\cdot(n-5)$
Is this nonsense or somewhat correct?