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A $2$-star is a tree with exactly two internal nodes. How many $2$-star in a complete graph are there?

In my opinion $${n\choose 2} \left(2^n-2\right),$$ my teacher said $${n\choose 2}2^{n-1}.$$

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    In your _opinion_? This is not a matter of opinion; it is either right or wrong. Do you have an _argument_ for your answer? – 2017-02-26
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    I have made some fast calculations, this is why I have chosen the word opinion. I have not written a proof. BTW, I just realised that what I proposed is wrong, now I will try again – 2017-02-26

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I get neither of these but instead $$ \binom n2 (3^{n-2}-2^{n-1}+1) $$ First a factor of $\binom n2$ for how to choose the two internal nodes.

Once they are chosen, each of the $n-2$ remaining nodes can be neighbors of either one of the internal nodes, or the other, or of neither -- except that we need to exclude choices where one of the nodes have no neighbors at all (and so is not internal).

If you're asking how many of the 2-stars use all of the nodes in the complete graph, a similar analysis leads to $$ \binom n2 (2^{n-2}-2) $$ which again matches neither of the proposals.

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    Thank you, I meant this second answer and I was just editing my "proposal" with the formula you gave, but you have been faster than me! – 2017-02-26