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?

Here's another way to think about it.
If everyone shook hands with everyone you'd have $\frac {17*16}2 = 136$ handshakes (divide by 2 because the above is double counting A shaking hands with B and so forth).
However, there are $\frac {17*2}2 = 17$ handshakes that aren't happening because neighbors aren't shaking hands (2 neighbors for each of the 17 people, again divided by 2 to remove duplicates).
So $136-17 = 119$
Pew's response is a little more direct, but finding the number by calculating the total possibilities minus the "not allowed" interactions is sometimes a little more intuitive for some people.
