Combinations: How many handshakes? I need help with the following question which I cannot seem to solve:
17 students are sitting in a circle. Each person shakes hands with everyone but his/her neighbours. How many handshakes have been exchanged?
My approach: no. of ways $ = 1 + 2 + 3 + ... + 14 = 7(15) = 105 $.
Apparently my answer is wrong (correct ans is 119). But I can't seem to understand why. Could someone please explain?

You can think of this in graph theory as the following:

$G$ has $17$ vertices with degree $14$, since there are no loops(people self handshaking), and no vertices are adjacent to the vertices beside them. So total degree is $17\times 14$.
We know that degree is equal to $2\times\text{number of edges}$ and hence there are $\frac{17\times 14}{2}=119$ edges in total, where edges represent handshakes.
