This is funny, my class just went over this.
Connect all the vertices. Tell me, if there are 5 vertices, how many line segments touch each vertex? the answer is 4.
If there are 6 vertices, then the answer is 5.
If there are 7, then the answer is 6.
Do you understand what I did (I have not answered the problem yet, I've merely gone through the first, most important part)? If not, then I will go into more detail...