Expected number of vertices for a given random graph

Question

So I have to find out the expected value of the number of vertices of degree 1. Let's the given graph if G(n,p). Then the $$\mathbb{P}(deg(v)=1) = {n \choose 1} (p)^{1}(1-p)^{n-1}$$

And $$\mathbb{E}X = (n){n\choose 1} p(1-p)^{n-1} = n^2p(1-p)^{n-1}$$

So I do not know if I am doing it right. Can one simplify it further? I am stuck here.

Thanks in advance.

Answer 1

You've made one mistake. There are only $n-1$ possible neighbors of a given vertex. So you should have $$n(n-1)p(1-p)^{n-2}$$